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If alpha and beta are the roots of the equation 2x2 − 4x + 7 = 0, find alpha2 + beta2.

March 28, 2026Mathematics

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Answer

-3

4 steps

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 4x + 7 = 0$ , find $\alpha^2 + \beta^2$ .

Identify the coefficients.
For the quadratic equation $ax^2 + bx + c = 0$ , we have $a = 2$ , $b = -4$ , $c = 7$ .
Find the sum and product of the roots.
Sum of roots: $\alpha + \beta = -\frac{b}{a}$
$\alpha + \beta = -\frac{-4}{2} = \frac{4}{2} = 2$
Product of roots: $\alpha \beta = \frac{c}{a}$
$\alpha \beta = \frac{7}{2}$
Use the identity for $\alpha^2 + \beta^2$ .
We know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$ .
Substitute the values of $\alpha + \beta$ and $\alpha \beta$ into the identity.
$\alpha^2 + \beta^2 = (2)^2 - 2\left(\frac{7}{2}\right)$ $\alpha^2 + \beta^2 = 4 - 7$ $\alpha^2 + \beta^2 = -3$
The value of $\alpha^2 + \beta^2$ is $\boxed{-3}$ .