An algebra problem by Ritik Devnani

Algebra Level 3

If a 2 − a − 10 = 0 a^2 - a - 10 = 0 , what is ( a + 1 ) ( a + 2 ) ( a − 4 ) = ? (a + 1) (a + 2) (a - 4) = ?


The answer is -8.

Ritik Devnani by Ritik Devnani , 17, India

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2 solutions

Chew-Seong Cheong
Aug 25, 2017

Given that a 2 − a − 10 = 0 a^2-a-10 = 0 ⟹ a 2 = a + 10 \implies \color{#3D99F6} a^2 = a + 10 . Now, we have:

( a + 1 ) ( a + 2 ) ( a − 4 ) = ( a 2 + 3 a + 2 ) ( a − 4 ) = ( a + 10 + 3 a + 2 ) ( a − 4 ) = ( 4 a + 12 ) ( a − 4 ) = 4 ( a + 3 ) ( a − 4 ) = 4 ( a 2 − a − 12 ) = 4 ( a 2 − a − 10 − 2 ) Note that a 2 − a − 10 = 0 = 4 ( 0 − 2 ) = − 8 \begin{aligned} (a+1)(a+2)(a-4) & = ({\color{#3D99F6} a^2} + 3a+2)(a-4) \\ & = ({\color{#3D99F6} a+10} + 3a+2)(a-4) \\ & = (4a+12)(a-4) \\ & = 4(a+3)(a-4) \\ & = 4(a^2 - a - 12) \\ & = 4({\color{#3D99F6}a^2 - a - 10} - 2) & \small \color{#3D99F6} \text{Note that } a^2-a-10 = 0 \\ & = 4({\color{#3D99F6}0} - 2) \\ & = \boxed{-8} \end{aligned}

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Thank You1 @Chew-Seong Cheong

Ritik Devnani - 3 years, 9 months ago
James Wilson
Jan 13, 2021

I expanded ( a + 1 ) ( a + 2 ) ( a − 4 ) (a+1)(a+2)(a-4) as a 3 − a 2 − 10 a − 8 = a ( a 2 − a − 10 ) − 8 = a ( 0 ) − 8 = − 8. a^3-a^2-10a-8=a(a^2-a-10)-8=a(0)-8=-8.

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