Question

If $\alpha \text{and}\beta$ are the the roots of $x^2-ax+b^2=0$, then ${\alpha }^2+{\beta }^2$ is equal to

• A. $2a^2-b^2$
• B. $a^2-b^2$
• C. $a^2+b^2$
• D. $a^2-2b^2$

Answer 1

The correct option is D $a^2-2b^2$

Given: $\alpha \text{and}\beta$ are the the roots of $x^2-ax+b^2=0$,

Using the relation between roots and coefficients of a quadratic equation, we get
${(\alpha +\beta )}^2={\left(\frac{a}{1}\right)}^2\Rightarrow {\alpha }^2+{\beta }^2+2\alpha \beta =\frac{a^2}{1^2}=a^2\Rightarrow {\alpha }^2+{\beta }^2+2\times \frac{b^2}{1}=a^2\Rightarrow {\alpha }^2+{\beta }^2=a^2-2b^2$