Simple Substitution

Algebra Level 3

6 ( x + 1 ) [ − 2 x 3 + 19 x 2 − 36 x ] \large 6(x+1) [-2x^{3} + 19x^{2} - 36x]

Find the value of the expression above if x 2 − 6 x = 6 \large x^{2} - 6x = 6 .


The answer is 36.

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Efren Medallo by Efren Medallo , 26, Philippines

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2 solutions

Chew-Seong Cheong
Dec 25, 2015

6 ( x + 1 ) ( − 2 x 3 + 19 x 2 − 36 x ) = 6 ( x + 1 ) ( − 2 x ( x 2 − 6 x − 6 ) + 7 x 2 − 48 x ) Note that x 2 − 6 x − 6 = 0 = 6 ( x + 1 ) ( − 0 + 7 ( x 2 − 6 x ) − 6 x ) Note that x 2 − 6 x = 6 = 6 ( x + 1 ) ( 42 − 6 x ) = 36 ( x + 1 ) ( 7 − x ) = 36 ( − x 2 + 6 x + 7 ) = 36 ( − 6 + 7 ) = 36 \begin{aligned} 6(x+1)(-2x^3+19x^2-36x) & = 6(x+1)(-2x(\color{#3D99F6}{x^2-6x-6})+7x^2-48x) \quad \quad \space \small \color{#3D99F6}{\text{Note that }x^2-6x-6=0} \\ & = 6(x+1)(-\color{#3D99F6}{0}+7(\color{#D61F06}{x^2-6x})-6x) \quad \quad \quad \quad \quad \quad \small \color{#D61F06}{\text{Note that }x^2-6x=6} \\ & = 6(x+1)(\color{#D61F06}{42}-6x) \\ & = 36(x+1)(7-x) \\ & = 36(\color{#D61F06}{-x^2+6x}+7) \\ & = 36(\color{#D61F06}{-6}+7) \\ & = \boxed{36} \end{aligned}

5 Helpful 0 Interesting 0 Brilliant 0 Confused
James Cochrane
May 27, 2015

From the last piece of information,

x 2 − 6 x − 6 = 0 x^2-6x-6=0

x = 3 + 15 x=3+\sqrt{15} or 3 − 15 3-\sqrt{15}

I assigned these values to letters on my calculator to save time and then plugged the letters into the expression equalling 36.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

but the better way is to manipulate the given polynomials

Devendra Singh - 5 years, 10 months ago

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