Question

If α, β are the roots of the equation 2x2 – 2x + 3 = 0, then form equation whose roots are α2 + 2 and β2 + 2.
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Answer 1

Given: $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^{2}-2x+3=0$.

To find: The quadratic equation whose roots are $({\alpha}^{2}+2)$ and $({\beta}^{2}+2)$.

Solution:

Since $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^{2}-2x+3=0$, by the relation between roots and coefficients, we get

$\quad\quad\color{blue}\alpha+\beta=-\frac{-2}{2}=1$

$\quad\quad\color{blue}\alpha\beta=\frac{3}{2}$

$\therefore{\alpha}^{2}+{\beta}^{2}$

$=(\alpha+\beta)^{2}-2\alpha\beta$

$=1^{2}-2\left(\frac{3}{2}\right)$

$=1-3$

$=-2$

$\Rightarrow \color{blue}{\alpha}^{2}+{\beta}^{2}=-2$

The quadratic equation whose roots are $({\alpha}^{2}+2)$ and $({\beta}^{2}+2)$ is given by

$\quad \{x-({\alpha}^{2}+2)\}\{x-({\beta}^{2}+2)\}=0$

$\Rightarrow x^{2}-({\alpha}^{2}+{\beta}^{2}+4)x+\{{\alpha}^{2}{\beta}^{2}+2({\alpha}^{2}+{\beta}^{2})+4\}=0$

$\Rightarrow x^{2}-(-2+4)x+\{\left(\frac{3}{2}\right)^{2}+2(-2)+4\}=0$

$\Rightarrow x^{2}-2x+\frac{9}{4}=0$

$\Rightarrow 4x^{2}-8x+9=0$

Answer: The quadratic equation whose roots are $({\alpha}^{2}+2)$ and $({\beta}^{2}+2)$ is $\color{blue}4x^{2}-8x+9=0$.