Question

If alpha and beta are the roots of the equation 2x^2+5x-12=0 find the value of alpha^2+beta^2​

Answer 1

Answer as a fraction: 73/4

Answer in decimal form: 18.25

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Step-by-step explanation:

If alpha and beta are the roots of the quadratic ax^2+bx+c = 0, then we can say

• alpha+beta = -b/a
• alpha*beta = c/a

For more information refer to Vieta's Formulas. Focus on the quadratic case.

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Comparing 2x^2+5x-12 = 0 to ax^2+bx+c = 0, we have

• a = 2
• b = 5
• c = -12

Which leads us to

• alpha+beta = -b/a = -5/2 = -2.5
• alpha*beta = c/a = -12/2 = -6

Or in short

• alpha+beta = -2.5
• alpha*beta = -6

Let's square both sides of the first equation to get

alpha+beta = -2.5

(alpha+beta)^2 = (-2.5)^2

alpha^2 + 2*alpha*beta + beta^2 = 6.25

alpha^2 + 2*(-6) + beta^2 = 6.25

alpha^2 - 12 + beta^2 = 6.25

alpha^2 + beta^2 = 6.25 + 12

alpha^2 + beta^2 = 18.25 = 73/4

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Verification:

Use the quadratic formula to solve 2x^2+5x-12 = 0 to get the roots of x = -4 and x = 3/2 = 1.5

Those are the values of alpha and beta in either order.

alpha^2 + beta^2 = (-4)^2 + (1.5)^2 = 18.25 = 73/4

The answer is confirmed.