Does there exist a quadratic equation whose coefficients are rati… - Vidyadip

Question

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

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Answer

Yes, a quadratic equation having coefficients as rational number, has irrational roots.
For example, $2x^2 - 3x - 15 = 0$ has rational coefficients.
$D = b^2 - 4ac (a = 2, b = -3, c = -15)$
$\Rightarrow D = 129$
$\therefore\ \text{Roots are given by x}=\frac{-\text{b}\pm\sqrt{\text{b}^2-4\text{ac}}}{2\text{a}}$
$\Rightarrow\ \text{x}=\frac{-(-3)\pm\sqrt{129}}{2\times2}$
$\Rightarrow\ \text{x}=\frac{3\pm\sqrt{129}}{4}$
The roots are irrational as $\sqrt{129}$ is irrational.

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