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:
\[ \sqrt{3} x^2 + 10x + 7\sqrt{3} = 0 \]

Step 1: Identify coefficients
\[ a = \sqrt{3}, \quad b = 10, \quad c = 7\sqrt{3} \]

Step 2: Write quadratic formula
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Calculate discriminant
\[ D = b^2 - 4ac = 10^2 - 4 \times \sqrt{3} \times 7\sqrt{3} = 100 - 4 \times \sqrt{3} \times 7\sqrt{3} \] Note that \(\sqrt{3} \times \sqrt{3} = 3\), so
\[ D = 100 - 4 \times 7 \times 3 = 100 - 84 = 16 \]

Step 4: Calculate roots
\[ x = \frac{-10 \pm \sqrt{16}}{2 \times \sqrt{3}} = \frac{-10 \pm 4}{2 \sqrt{3}} \]
Two roots:
\[ x_1 = \frac{-10 + 4}{2 \sqrt{3}} = \frac{-6}{2 \sqrt{3}} = \frac{-3}{\sqrt{3}} = -\sqrt{3} \] \[ x_2 = \frac{-10 - 4}{2 \sqrt{3}} = \frac{-14}{2 \sqrt{3}} = \frac{-7}{\sqrt{3}} = -\frac{7 \sqrt{3}}{3} \]

Final Answer:
\[ x = -\sqrt{3} \quad \text{or} \quad x = -\frac{7 \sqrt{3}}{3} \]