Question

solve $3x^2 + 6x + 3 = 0$ using the quadratic formula.

$x = \square$

Explanation:

Step1: Recall Quadratic Formula

The quadratic formula for a quadratic equation \(ax^2 + bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). For the equation \(3x^2+6x + 3 = 0\), we identify \(a = 3\), \(b = 6\), and \(c = 3\).

Step2: Calculate the Discriminant

First, calculate the discriminant \(D=b^2-4ac\). Substitute \(a = 3\), \(b = 6\), \(c = 3\) into the formula: \(D=(6)^2-4\times3\times3=36 - 36=0\).

Step3: Substitute into Quadratic Formula

Now substitute \(a = 3\), \(b = 6\), and \(D = 0\) into the quadratic formula: \(x=\frac{-6\pm\sqrt{0}}{2\times3}=\frac{-6\pm0}{6}\).

Step4: Simplify the Expression

Simplify \(\frac{-6\pm0}{6}\). Since adding or subtracting 0 doesn't change the numerator, we have \(x=\frac{-6}{6}=- 1\). Both solutions (from the \(\pm\)) are the same because the discriminant is 0, so there is one real solution (a repeated root) of \(x=-1\).

Answer:

\(x = -1\)