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How To Solve Second Degree Equations By Factoring Or By Using The Quadratic Formula
Feb 19, 2025
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Welcome back to another math lecture. Today we will learn how to solve quadratic equations. Many students tend to get frightened when they hear words like quadratic. First we'll define what a quadratic equation is and then we'll learn how to solve it by factoring and via the quadratic formula
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0:00
hello and welcome back to another hour
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of mathematics okay today we're going to
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talk about quadratic equations
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i know people don't like quadratic
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equations because they tend to be
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a little bit confusing but we're going
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to try to uh
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get rid of that confusion because it's
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actually easy to deal with
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so first thing i want to do is i want to
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show you how to
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work with quadratic equations there's
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like multiple ways you can do it
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so we can do it by factoring we can do
0:25
it by using the quadratic formula
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but number one thing we need to
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understand is again definition
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what is a quadratic equation that's what
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we need to understand
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now any equation in the form
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a x squared plus b
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x plus c equals
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zero is called a quadratic equation
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okay and a x squared plus b x plus c
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is a polynomial of a second degree okay
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polynomial of a second degree because
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the leading degree here is two so this
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is a second degree
0:59
polynomial now this is a second degree
1:01
equation so
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a x squared plus b x plus c is equal to
1:05
zero is called
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a quadratic equation now how do you
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solve
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this quadratic equation there are
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multiple ways to do it
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number one i'm going to teach you how to
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do it by factoring
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okay now in this quadratic equation
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there are things that we need to know
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about
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those coefficients a b
1:25
and c those are important to to know all
1:27
right
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so now let me give you an example of a
1:30
quadratic equation and we're going to
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try to solve this
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so let's say i have x squared
1:36
minus um 14x
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plus 33 equals to zero
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and the question is solve for x
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so a lot of times you go i don't know
1:48
what to do and i was just like you two i
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didn't know what to do
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until i was in high school and my
1:53
teacher or
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actually in middle school and my teacher
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taught me how to do it
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by using the factoring so what i do here
2:01
is this
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i look at the coefficient okay the a
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this is a here this is b and this is c
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a is equal to 1 b is equal to negative
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14
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and c is equal to 33 since a is equal to
2:16
1
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um i can actually first what we're going
2:19
to do by by factoring so let's let's do
2:21
that here we have our coefficient
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i just wanted to show you what they were
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this is this qualifies as a
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second degree equation or quadratic
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equation so we're going to use the
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factoring so i'll do what
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add 1 times 33 a
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times c which is 1 multiplied by 33
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and that gives me 33. now the second
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step is
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you want to find two numbers whose
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product
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is 33 and sum is negative
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14. again i'll repeat that again you
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want to find two numbers
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who add up to negative 14 and whose
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multiplication gives you
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33. so first let's write
3:02
this out the product p we're going to
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call the product
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p has to give you 33 and the sum
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or the addition or subtraction
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gives you negative 14. so how do you
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find these numbers
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remember i told you yesterday you have
3:18
to know your multiplication table
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it does help a lot so you have to have
3:23
the basics the foundation if you don't
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know your multiplication table
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you're going to have a hard time so now
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let's break down 33.
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33 can be written as what 3 times
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11 right that's the easiest way to do it
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3 times 11.
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now look at this here 3 plus 11 gives me
3:40
14
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right if i do 3 plus 11 i get 14.
3:45
but i don't want 14 i want negative 14.
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so what do i do
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all i have to do is this i know that if
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i do negative 3
3:53
times negative 11 i still get what 33
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and if i do negative 3 minus 11
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i get negative 14 which is exactly what
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i want
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because i know that negative 3 plus
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negative 11
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gives me negative 14 when you add two
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negatives what do you do
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you add the numbers and you carry the
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sign in this case is a negative
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so i want negative 3 and negative 11. i
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have my two digits so if i know that
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negative 3
4:20
times negative 11 gives me 33 and i also
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know that negative 3
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plus negative 11 gives me negative 14.
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so i found
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my two numbers i'm pretty much done with
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this problem
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so all i have to do now is factor my
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second degree equations with coefficient
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a equals one now all i have to do just
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factor it right it's going to be x minus
4:42
3
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and then x minus 11 equals to 0
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and all i have to do now is set each one
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of those factors equal to 0 and solve
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for x so basically it's going to be x
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minus 3 equals to 0
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and x minus 11 equals to 0. so in this
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case we're going to have
5:00
x equals 3 because you add 3 on both
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sides
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right and then here x equals 11
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and this is my solution so my solutions
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are x
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equals 3 and x equals 11. that takes me
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to our next
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problem and where a is greater than 1
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and we're going to see
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what to do let's find an example let me
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write
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3x squared plus
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20x minus 7
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and now we want to again solve this
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equation equals to zero but the first
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step is always doing what we want to
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factor
5:39
this we want to factor it right so how
5:42
do we factor
5:43
we're going to use the method called uh
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factor
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by grouping we're going to group this
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bar we're going to repeat the same step
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again we need to know what
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a b and c are in this case
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a is 3 right
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a is 3 b
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is 20 and c
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is negative seven so i do a times c
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again it's the same method a times c
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so in this case a times c gives me three
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times negative seven and that gives me
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negative 21. so what did we say in the
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last problem
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so what we're going to do now is we want
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two numbers whose product
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p gives me negative 21 and sum
6:30
s has to give me 20. so how do i find
6:34
these numbers
6:35
well again if you know your
6:36
multiplication table it does help
6:38
so i know that 3 times 7
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gives me 21. i want negative 21. so
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either negative 7
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negative 3 times 7 equals negative 21 or
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negative 7 times 3
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that gives me negative 21. but if i do
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negative
6:55
3 plus 7 i get 4 and if i do negative 7
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plus 3 i get negative 4. so that doesn't
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work because my pro i want my my
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sum to give me 20. so let me try
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something else what about negative 20
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times one negative 20 times one gives me
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what
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it gives me uh no negative 21
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times 1 that gives me negative 21 right
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but then negative 21 plus 1 gives me
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negative 20. that's not good so all i
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have to do is just switch my
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my signs so i'm going to do 21 times
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negative 1
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that gives me negative 21 and the sum
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21 minus 1 gives me 20. so
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these two numbers are going to work
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because 21 times negative 1
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gives me negative 21 and then 21 minus 1
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gives me 20. so now that i have my two
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digits i'm going to rewrite
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this whole equation i'm going to rewrite
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it
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so this is going to turn into
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3x squared
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plus 21x minus
8:02
1x minus 7 equals to
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0. and then the next step is we're going
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to
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factor by grouping so i'm gonna do these
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two together
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and i'm gonna do these these two
8:13
together i need to find my gcf
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the gcf is what the greatest common
8:19
factor
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so what is the greatest common factor
8:21
between 3x square and 21x
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is 3x so if i plot 3x as a factor
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i'm left with x plus
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7 right now in this case
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what's my greatest common factor between
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negative 1 x and negative 7
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minus 7 is negative 1. so if i apply a
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negative 1
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i'm left with x plus 7.
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now look this is beautiful now i have my
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common factor
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i can just factor this right easy it
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turns into what x
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plus 7. times three x
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minus one equals to zero and then the
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next step is just to set each one of
9:00
those equal to zero and solve for x
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so let's see raise the top and then
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we're gonna solve this problem here
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so x plus seven equals to zero
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that means x equals equal to negative
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seven and then three
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x minus one equals to zero that means
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three x
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is equal to one and if you divide by
9:22
three
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x is equal to one third so my solutions
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are
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negative seven and one third and you are
9:29
done
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so this is called um factoring by
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grouping
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so we learned the simple factoring when
9:35
you when
9:36
a is equal to one and now when a is more
9:38
than one
9:39
this is how you do it by grouping now
9:42
let's say you don't like this factoring
9:44
you just don't like this kind of stuff
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but you just want to do it
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in a different way you want to do it uh
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you said to me time i just did this
9:52
factoring
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i don't like it i just despise it i want
9:55
to learn how to do it another way
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because some people love formulas some
9:59
people love
10:00
formulas right so we're going to use the
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quadratic formula
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and i've learned this again several
10:06
years ago the quadratic formula is
10:08
x equals negative
10:11
b plus or minus
10:14
square root of b squared minus 4 ac
10:19
over 2a again this is a ugly looking
10:23
formula i don't like using it i like
10:25
the factoring but some people don't like
10:27
the formula so i'm going to show you how
10:28
to use it
10:29
so let's go back to the same problems
10:31
that we had earlier the first one was
10:33
x squared minus 14x
10:37
plus 33 and then you want to use
10:40
this formula to solve the problem how do
10:42
you do it again you need to identify
10:44
a b and c in this case a
10:47
is 1 b is negative 14
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and c is 33. now all we have to do is
10:54
just plug it into the equation
10:55
into the formula so x is going to be
10:58
negative b
10:59
which is negative negative 14
11:02
right negative negative 14. my 14 looks
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like uh
11:07
my 14 looks almost like a 16 so i need
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to fix this
11:12
let me rewrite the equation is x squared
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minus 14 x plus 33
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and x is negative b b is negative 14 so
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negative b
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will be what will be negative negative
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14 right
11:28
plus or minus b squared which is
11:31
negative 14 squared
11:34
all right minus 4 times ac
11:38
minus 4 times a is 1
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c is 33 you see
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we end up with a gigantic looking thing
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over 2a 2 times 1.
11:48
now we need to find negative 14 squared
11:51
14 squared
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and i have to use my calculator 14
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squared will be
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let's do that on the side since i don't
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have my calculator i'm just going to do
11:59
it by hand
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14 squared let me multiply 14 in front
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of you
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i didn't want to do this but i have two
12:06
four and four is sixteen
12:07
four and one is four fifty 56 here and
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then we have
12:10
1 and 4 1 2 6
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9 196 okay so we're going to put it in
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step by step so we have x
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equals positive 14 plus or minus
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196 minus 4 times 33
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let's do 4 times 33 on the side
12:33
4 and 3 is 12 4 3 squared 132
12:36
minus 132 over 2
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all right and then the next step is
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x equals 14 plus or minus
12:48
196 minus 132 that is uh four here
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and three and nine that's 664
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over two now i don't know if you were
12:58
listening to my
12:59
lecture yesterday we talked about uh
13:02
perfect
13:02
squares okay now 64 is a perfect
13:06
square so that's going to be easy square
13:08
root of 64
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is 8. and i talked about what square
13:11
root of 64
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is it means one number multiplied by
13:15
itself
13:16
gives you uh 64. and this is 8 so square
13:19
root of 64 is 8. so i'm going to erase
13:21
the top part
13:22
and we're going to find our solution so
13:24
this turns into x
13:25
equals 14 plus or minus
13:29
8 over 2 and now we have to split
13:32
our answers so the first one is going to
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be x equals
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14 plus 8 over 2
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or x equals 14 minus 8
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over 2. our first solution is 14 plus 8
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that's 22
13:48
22 by 2 and the last one is x equals 14
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minus 8
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that is 6 over 2 so we have x equals 11
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or x equals 3. now this is the factoring
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this is the
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finding the solutions by using the
14:02
quadratic formula
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now this is a long process as you can
14:06
see with the factoring is much easier
14:08
with the factoring all i have to find
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was
14:10
two numbers whose product is 33 and sum
14:13
is 14
14:13
and when we did it we had negative 3 and
14:16
negative 11
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and then we did our factoring in this
14:18
case it's just too long and i don't want
14:20
to waste my time doing all these things
14:22
this is just way too long right so i
14:24
hope you enjoyed this
14:25
and i'm looking forward to another um
14:29
one of these all right again thank you
14:31
and if you need tutoring
14:33
please don't hesitate to reach out to me
14:35
i'll have a link for in the description
14:36
all right thanks for watching and stay
14:38
tuned for more stuff
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