Math Problem Solver

The quadratic equation \( 3 {x}^{2} + 12 x + 5 = 0 \) has two solutions: \[ x = -2 + \frac{\sqrt{21}}{3} , \quad x = -2 - \frac{\sqrt{21}}{3} \]

---

The quadratic equation in standard form is: \[ 3 x^{2} + 12 x + 5 = 0 \]

Subtract the constant term \( 5 \) from both sides of the equation: \[ 3 x^{2} + 12 x = -5 \]

Divide both sides of the equation by the quadratic coefficient \( 3 \), so the quadratic coefficient on the left hand side of the resulting equation is \(1\): \[ x^{2} + 4 x = - \frac{5}{3} \]

Complete the square on the left hand side by adding the square of half the linear coefficient, \( 4 \), to both sides of the equation: \[ x^{2} + 4 x + 4 = \frac{7}{3} \]

Rewrite the left hand side of the equation as a square: \[ \left(x + 2\right)^{2} = \frac{7}{3} \]

Rewrite the right hand side as a square, and subtract from both sides: \[ \left(x + 2\right)^{2} - \left(\frac{\sqrt{21}}{3}\right)^{2} = 0 \]

Use the identity \( a^2 - b^2 = (a-b)(a+b) \) to rewrite the difference of squared on the left hand side as a product: \[ \left(\left(x + 2\right) - \frac{\sqrt{21}}{3}\right) \left(\left(x + 2\right) + \frac{\sqrt{21}}{3}\right) = 0 \]

\[ x = -2 + \frac{\sqrt{21}}{3} , \quad x = -2 - \frac{\sqrt{21}}{3} \]