Question

Solve the quadratic equation \(\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0\) using the quadratic formula.

Hint:

Always simplify the discriminant carefully when irrational numbers like \(\sqrt{3}\) are involved.

Answer 1

Given quadratic equation: \(\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0\)
Compare with \(ax^2 + bx + c = 0\)
Here, \(a = \sqrt{3}, b = 10, c = 7\sqrt{3}\) Using quadratic formula: \[ x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \Rightarrow x = \dfrac{-10 \pm \sqrt{(10)^2 - 4(\sqrt{3})(7\sqrt{3})}}{2\sqrt{3}} \] \[ = \dfrac{-10 \pm \sqrt{100 - 4 \cdot \sqrt{3} \cdot 7\sqrt{3}}}{2\sqrt{3}} = \dfrac{-10 \pm \sqrt{100 - 84}}{2\sqrt{3}} = \dfrac{-10 \pm \sqrt{16}}{2\sqrt{3}} = \dfrac{-10 \pm 4}{2\sqrt{3}} \] Therefore: \[ x_1 = \dfrac{-10 + 4}{2\sqrt{3}} = \dfrac{-6}{2\sqrt{3}} = \dfrac{-3}{\sqrt{3}} = -\sqrt{3}, \quad x_2 = \dfrac{-10 - 4}{2\sqrt{3}} = \dfrac{-14}{2\sqrt{3}} = \dfrac{-7}{\sqrt{3}} \]