79th Problem 2016

Algebra Level 2

Add the sum and product of the roots of the following equation:

x 2 + 2 x − 6 = 2 { x }^{ 2 }+2x-6=2

Check out the set: 2016 Problems


The answer is -10.

Angela Fajardo by Angela Fajardo , 19, Philippines

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2 solutions

x 2 + 2 x − 6 = 2 x^2 + 2x - 6 = 2 ⟹ \implies x 2 + 2 x − 8 x^2 + 2x - 8

By Vieta's Formula ,

The sum of the roots is

x 1 + x 2 = − B A = − 2 1 = − 2 x_1 + x_2 = \frac{-B}{A} = \frac{-2}{1} = -2

The product of the roots is

x 1 x 2 = C A = − 8 1 = − 8 x_1x_2 = \frac{C}{A} = \frac{-8}{1} = -8

Adding the sum and the product of the roots, we get

( x 1 + x 2 ) + x 1 x 2 = − 2 + ( − 8 ) = (x_1 + x_2) + x_1x_2 = -2 + (-8) = − 10 \boxed{-10}

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Hung Woei Neoh
Jun 10, 2016

x 2 + 2 x − 6 = 2 x 2 + 2 x − 8 = 0 x^2+2x-6=2\\ x^2+2x-8=0

Now, you should know that:

x 2 − ( Sum of roots ) x + ( Product of roots ) = 0 x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0

Therefore,

Sum of roots + Product of roots = − 2 − 8 = − 10 = -2-8=\boxed{-10}

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