# How to Use the Quadratic Formula

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A quadratic equation is one that contains a single variable and in which the variable is squared. The standard form for this type of equation, which always produces a parabola when graphed, is ​ax2 + ​bx​ + ​c​ = 0, where ​a​, ​b​ and ​c​ are constants. Finding solutions isn't as straightforward as it is for a linear equation, and part of the reason is that, because of the squared term, there are always two solutions. You can use one of three methods to solve a quadratic equation. You can factor the terms, which works best with simpler equations, or you can complete the square. The third method is to use the quadratic formula, which a generalized solution to every quadratic equation.

For a general quadratic equation of the form ​ax2 + ​bx​ + ​c​ = 0, the solutions are given by this formula:

x = \frac{−b ±\sqrt{b^2 − 4ac} }{2a}

Note that the ± sign inside the brackets means that there are always two solutions. One of the solutions uses

\frac{−b +\sqrt{b^2 − 4ac} }{2a}

and the other solution uses

\frac{−b -\sqrt{b^2 − 4ac} }{2a}

Before you can make use of the quadratic formula, you have to make sure the equation is in standard form. It may not be. Some ​x2 terms may be on both sides of the equation, so you'll have to collect those on the right side. Do the same with all x terms and constants.

Example: Find the solutions to the equation

3x^2 - 12 = 2x(x -1)

Expand the brackets:

3x^2 - 12 = 2x^2 - 2x

Subtract 2​x2 and from both sides. Add 2​x​ to both sides

3x^2 - 2x^2 + 2x - 12 = 2x^2 -2x^2 -2x + 2x \\ 3x^2 - 2x^2 + 2x - 12 = 0 \\ x^2 - 2x -12 = 0

This equation is in standard form ​ax2 + ​bx​ + ​c​ = 0 where ​a​ = 1, ​b​ = −2 and ​c​ = 12