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solve the equation x ra 9th power - x
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Cub = 6 as usual pause the video before
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watching further and try to solve it on
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your own if there's an opportunity to
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simplify the process we should take
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advantage of it therefore in this case
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let's try to make it easier by
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substitution for example let's denote X
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cubed as T then x raed to the 9th power
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can be written as X cubed all Rays to
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the thir power - x Cub let's write it
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this way and move the number six
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immediately to the left side with a
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negative sign now we can substitute the
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new variable T in place of X cubed as a
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result we get T Cub - t - 6 = 0 this
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equation can be solved in various ways
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for example we could try to find one of
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the roots among the divisors of the
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constant term however we'll take a
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different approach we'll try to factor
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grouping to do this we'll add and
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subtract T cubed using the difference or
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sum of cubes for example from the number
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-6 on the left side of the equation
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we'll subtract two but to ensure nothing
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changes we'll also add two then we'll
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group8 with t cubed and leave 2 with - t
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so on the left side of the equation
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we'll have t Cub - 8 - t + 2 all
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equaling 0 8 is 2 cubed so we've now
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obtained the necessary difference of
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Cubes T Cub - 2 Cub we still have - t
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and -2 and all of this equals z let's
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expand the difference of Cubes using the
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standard formula we'll get the
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difference of these numbers T minus 2
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multiplied by the incomplete square of
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their sum that is ultied T ^2 + 2 t + 4
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we still have t and 2 and all of this
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must equal 0 tus 2 can be factored out
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inside the parentheses we are left with
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t ^2 + 2 t + 4 - 1 and all of this must
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equal zero so we finally have t - 2 *
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t^2 + 2 t + 3 equaling 0 the product of
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two polinomial equals 0 when at least
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one of them equals z however since the
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discriminant of the quadratic trinomial
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is less than zero the second polinomial
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has no real Roots therefore the left
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side of our equation equals 0 only when
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the first Factor T minus 2 equal 0 from
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this T must equal 2 returning to the
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original variable we find that X cubed
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must equal 2 from which x equal the cube
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root of 2 write the answer the equation
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has a unique solution X must equal the
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cube root of two the problem is solved
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if you understood the solution give give
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a thumbs up leave a comment and don't
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