The roots of the quadratic equation x^2 - 2sq | vedclass

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Question

(C) For the quadratic equation $ax^2 + bx + c = 0$, the discriminant $D$ is given by $D = b^2 - 4ac$. Here, $a = 1$, $b = -2\sqrt{3}$, and $c = -22$.

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real, irrational and unequal

Vedclass · Answer

(C) For the quadratic equation $ax^2 + bx + c = 0$, the discriminant $D$ is given by $D = b^2 - 4ac$. Here, $a = 1$, $b = -2\sqrt{3}$, and $c = -22$. $D = (-2\sqrt{3})^2 - 4(1)(-22) = 12 + 88 = 100$. Since $D > 0$, the roots are real and unequal. Using the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$: $x = \frac{2\sqrt{3} \pm \sqrt{100}}{2} = \frac{2\sqrt{3} \pm 10}{2} = \sqrt{3} \pm 5$. Since $\sqrt{3}$ is an irrational number, the roots $\sqrt{3} + 5$ and $\sqrt{3} - 5$ are irrational. Therefore, the roots are real, irrational, and unequal.

The roots of the quadratic equation $x^2 - 2\sqrt{3}x - 22 = 0$ are:

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