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welcome to Mathematics and Sciences
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for more information click the link in
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the description Today's tutorial video
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is the continuation of the tangent
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question Hence we are now supposed to
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determine the second point of
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intersection If you missed out on part
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one of this video click the link in the
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description specified as part one of the
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video So let's jump straight into it The
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question reads as follows Determine the
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point of the point where the tangent
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line calculated on part one of the video
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intersects the curve or the function g
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of x for the second time Finding the
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point of intersection we have to equate
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the two functions g ofx and f ofx The
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tangent line function given as -8x + 34
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= to -x cub - 2x^2 + 11 x + 12 Transpose
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everything to the left hand side of the
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equal sign and simplify by grouping of
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like terms This implies x cub + 2x^2 -
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19 x + 22 Secondly we have to determine
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the factors of the constant 22 as 1 and
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22 2 and 11 These factors we then be
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rewritten as 1 and minus1 22 and -22 2
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Remember we already have the point of
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tangency at x= 22 Hence we know that at
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x= 22 is the point of intersection of
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function f and g Thus f ofx is equals to
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g ofx Proving that even further we can
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substitute x = 2 2 into x cub + 2x^2 -
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19 x + 22 We are finding x = 2 to be a
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factor of x cub + 2x^2 - 19 x + 22 Using
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long division finding our divisor will
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be using x = 2 transpose 2 to the left
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of the equal sign giving us x - 2 x cub
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+ 2x^2 - 19 x + 22 will be our dividend
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x cubed will be divided by x from the
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divisor xus 2 giving us x^2 as quotient
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The quotient x^2 will then multiply x -
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2 our divisor giving us the new dividend
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x cub - 2x^2 Then we subtract the two
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dividend giving us 4x^2 from the
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dividend 4x^2 and x from the divisor x -
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2 giving us 4x as the quotient Bring
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down -19x giving us our new dividend
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4x^2 - 19x The quotient for x multiplied
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by the divisor x - 2 gives us 4x^2 - 8x
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as our second dividend The difference
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between these two dividends dividend for
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x^2 - 19x and 4x^2 - 8x gives us -1x
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divide 11x from the x in the divisor x -
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2 giving us the quotient -1 Bring down
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22 giving us our new dividend -1x + 22
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The quotient -11 multiplied by the
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divisor x - 2 gives us -1x + 22 as our
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second dividend The difference between
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these two dividends -1x + 22 and
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negative 11x + 22 give us zero Therefore
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our new quotient is x^2 + 4x - 11 Since
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we determined the quotient calculated
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from long division giving us x^2 + 4x -1
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will the use the quadratics formula
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given as x= to b plus or minus into the
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square roo of b ^ 2 minus for a c
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all to a from our quotient text^ 2 + 4
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for x -1 is = 21 b is equ= 2 4 and c =
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substitute the values of a b and c into
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the quadratic formula giving us -4 plus
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or minus of 60 all over which can
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be also simplified to -2 plus or
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minus of 15 Since after find the
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xcoordinate of the second point of
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intersection we can substitute our
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xcoordinate into either f ofx or g ofx I
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chose f ofx but you can also choose g of
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x2 Substituting the x value into f ofx
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given as f of the x value -4 plus or
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minus of 60 all over to8 into -4
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plus or minus 60 all over + 34
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giving Thus the y-coordinate of 18 minus
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or + 8 square roo of 15 Therefore
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point of points of intersection of f ofx
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and g ofx are calculated Thanks for
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toning into the math and science academy
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tutorial session Don't to forget to
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comment for any inquiries Like if you
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enjoyed the video Subscribe for more
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you so much and let's keep practicing
