# Solving the quadratic equation with the quadratic formula

Solve the quadratic equation with the quadratic formula

The method of completing the square can be used to develop a formula that enables us to solve any quadratic equation. Consider the general quadratic equation :

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By applying the method of completing the square, we can obtain the solutions in terms of the coefficients *a*, *b, *and *c. *

Since the coefficient of x is *b/a*, the square of one – half the coefficient of x is to each side.

We now have a formula for solving any quadratic equation. It is usually stated as follows :

**The Quadratic Formula**

The solutions of the quadratic equation

Are given by

**Example 1 :**

Solve

Using the quadratic formula.

Solution :

We first put the equation in standard form in order to identify *a, b, *and *c. *

*a = 3, b = – 4, c = 1.*

Substituting these values for *a, b, *and *c* into the quadratic formula, we have :

Therefore,

The solution are 1 and 1/3.

**Example 2 # :**

Solve

Using the quadratic formula.

Solution :

*a = *1, *b = *1, *c = *-1

The solution are

**Example 3 # :**

Solve

Using the quadratic formula.

Solution :

We first simplify the equation by multiplying both sides by the lcd 6.

*a = 1, b = 2, c = – 2*

Dividing the denominator 2 into both terms in the numerator simplifies the solution to

The solutions are

**Example 4# :**

Solve

Using the quadratic formula.

Solution :

*a = 1, b = -7 , c = 0*

x = 7 or x = 0

The solutions are 7 and 0.

**Example 5# :**

Solve

Using the quadratic formula.

Solution :

*a = 1, b = 2, c = 5*

But is not a real number. Therefore this quadratic equation has no real number solution.

The method of solving a quadratic equation by completing the square can be rather tedious, and for that reason it is seldom used to solve a quadratic equation. The operation of completing the square does, however, have a variety of other application in algebra.