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The Quadratic Formula is a widely used mathematical formula to solve quadratic equations. This formula can be expressed in the following ways: \begin{equation*} x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{equation*}

The quadratic formula helps us to find solutions to equations of the form ax²+bx+c=0.

### Example 1

For example, given the equation 3x² + 12x + 9 = 0, the quadratic formula can be used to find the two solutions.

We can apply the formula as follows:

\begin{equation*} a = 3, b = 12, c = 9 \end{equation*} \begin{equation*} x = \frac{-12 \pm \sqrt{144-4(3)(9)}}{2(3)} \end{equation*} \begin{equation*} x = \frac{-12 \pm \sqrt{48}}{6} \end{equation*} \begin{equation*} x = \frac{-12 \pm 2 \sqrt{6}}{6} \end{equation*} The two solutions for the equation 3x² + 12x + 9 = 0 are therefore \begin{equation*} x = -2 - \sqrt{6} \end{equation*} \begin{equation*} x = -2 + \sqrt{6} \end{equation*}

### Examples

Here are 4 examples of when we can use the quadratic formula:

• To solve equations of the form ax²+bx+c=0.
• To calculate the roots of a polynomial.
• To calculate the zeroes of a parabola.
• To solve questions in algebra, calculus and other branches of mathematics.