The Quadratic Formula

You can solve any quadratic equation by completing the square.

Now we will complete the square to solve ax² + bx + c = 0. The solutions will be expressed in terms of a, b, and c. These solutions will give us a formula we can use to solve any quadratic equation.

Step 1 Isolate the x2-term and the x-term on one side of the equation. Subtract c from both sides of the equation.	ax2 + bx + c = 0 ax2 + bx = -c
Step 2 If the coefficient of x2 is not 1, divide both sides of the equation by the coefficient of x2. The coefficient of x2 is a.
Divide both sides of the equation by a.
Step 3 Find the number that completes the square: Multiply the coefficient of x by . Square the result. The coefficient of the x-term is . Step 4 Add the result of Step 3 to both sides of the equation.
Add to both sides of the equation. To combine like terms on the right side, write both fractions with denominator 4a2.
Combine like terms on the right side. In the numerator, write the b2-term first. Step 5 Write the trinomial as the square of a binomial. Step 6 Finish solving using the Square Root Property.
Use the Square Root Property. Rather than writing two separate equations, we write a single equation using the ± sign.
Subtract from both sides and simplify the radical.
Combine the fractions into a single fraction.

Note:

If a > 0, then 4a² = 2a.

If a < 0, then 4a² = -2a.

So,

The result is called the quadratic formula.

Formula — The Quadratic Formula

The solutions of the quadratic equation ax² + bx + c = 0 are given by the quadratic formula:

Here, a, b, and c are real numbers and a ≠ 0.