How To Use The Quadratic Formula

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 ​ax​² + ​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 ​ax​² + ​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 ​x​² 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

\(3x^2 – 12 = 2x(x -1)\)

1. Convert to standard form

Expand the brackets:

\(3x^2 – 12 = 2x^2 – 2x\)

Subtract 2​x​² 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 ​ax​² + ​bx​ + ​c​ = 0 where ​a​ = 1, ​b​ = −2 and ​c​ = 12

2. Plug the values of a, b and c into the quadratic formula

The quadratic formula is

\(x = \frac{−b ±\sqrt{b^2 − 4ac} }{2a}\)

Since ​a​ = 1, ​b​ = −2 and ​c​ = −12, this becomes

\(x = \frac{−(-2) ±\sqrt{(-2)^2 − 4×1×(-12)} }{2×1}\)

3. Simplify

\(x = \frac{2 ±\sqrt{(4+ 48} }{2} \
\,\
x = \frac{2 ±\sqrt{52} }{2} \
\,\
x = \frac{2 ±7.21 }{2} \
\,\
x = \frac{9.21}{2} \text{ and } x = \frac{−5.21}{2} \
\,\
x = 4.605 \text{ and } x = −2.605\)

You can solve quadratic equations by factoring. To do this, you more or less guess at a pair of numbers that, when added together, give the constant ​b​ and, when multiplied together, give the constant ​c​. This method can be difficult when fractions are involved. and wouldn't work well for the above example.

The other method is to complete the square. If you have an equation is standard form, ​ax​² + ​bx​ + ​c​ = 0, put ​c​ on the right side and add the term (​b​/2)² to both sides. This allows you to express the left side as (​x​ + ​d​)², where ​d​ is a constant. You can then take the square root of both sides and solve for ​x​. Again, the equation in the above example is easier to solve using the quadratic formula.