The roots of the quadratic equation $x^2+x-3=0$ are $\alpha$ and $\beta$. Write down the value

Question

The roots of the quadratic equation $x^2+x-3=0$ are $\alpha$ and $\beta$. Write down the values of $\alpha+\beta$ and of $\alpha\beta$. Without solving the equation, find the value of each of the following.$\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}$ $\left(\alpha-\dfrac{2}{\beta^2}\right)\left(\beta-\dfrac{2}{\alpha^2}\right)$ Determine the quadratic equation with integer coefficients which has roots $\alpha-\dfrac{2}{\beta^2}$ and $\beta-\dfrac{2}{\alpha^2}$.

Accepted Answer

(a) $-1,\ -3$ (b)(i) $\frac{7}{9}$ (ii) $-\frac{29}{9}$ (c) $9x^2+23x-29=0$

Reciprocal-square evaluation leading to a new equation

Reciprocal-square evaluation leading to a new equation

Solution:

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