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Algebra Level 3

Find the sum of the values of x x for which the roots g g and h h of the equation t 2 − 8 t + x = 0 t^2 - 8t +x = 0 satisfy the conditions that g 2 + h 2 = 4 g^2 + h^2 = 4 .


The answer is 30.

Shreya R by Shreya R , 21, India

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2 solutions

for t^2+bt+c. sum of roots=-(b/a) and product=(c/a). g+h=8 and gh=x. (g+h)^2=g^2+h^2+2gh. =>64=4+2x. 2x=60=> x=30.

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@Deeksha Maheshwari How do you know that this is the only value for x x

Abdur Rehman Zahid - 6 years, 5 months ago
Jesse Nieminen
Sep 16, 2016

Using Vieta's formula we know that g + h = 8 g+h=8 and g h = x gh=x .

Now ( g + h ) 2 − 2 g h = g 2 + h 2 = 4 ⟹ 8 2 − 2 x = 4 ⟹ x = 30 \left(g+h\right)^2-2gh=g^2+h^2=4\implies8^2-2x=4\implies x=30 .

Hence, 30 30 is the only possible value for x x , and thus the solution is 30 \boxed{30}

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