Problem 130 - College Algebra

Question

Solve each equation by the method of your choice. $$\sqrt{3} x^{2}+6 x+7 \sqrt{3}=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -1 + 2i\sqrt{3}\) and \(x = -1 - 2i\sqrt{3}\).

Step 1: Identify the coefficients

The coefficients in the given equation are \(a = \sqrt{3}\), \(b = 6\), and \(c = 7\sqrt{3}\)

Step 2: Substitute the coefficients into the quadratic formula

Substitute \(a = \sqrt{3}\), \(b = 6\), and \(c = 7\sqrt{3}\) into the quadratic formula. This results in, \[x = \frac{-6 \pm \sqrt{(6)^{2} - 4(\sqrt{3})(7\sqrt{3})}}{2\sqrt{3}}\]

Step 3: Calculate the discriminant

Calculate the discriminant \(b^{2} - 4ac = (6)^{2} - 4(\sqrt{3})(7\sqrt{3})\). This simplifies to 36- 84 = -48. Since the discriminant is less than zero, the equation has two complex solutions.

Step 4: Calculate the roots

Substitute the discriminant into the formula we have, \[x = \frac{-6 \pm \sqrt{-48}}{2\sqrt{3}}\], which simplifies to \[x = -1 \pm 2i\sqrt{3}\]