You can solve any quadratic equation by completing the square.
Now we will complete the square to solve ax^{2} + 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 x^{2}term and the xterm
on one side of the equation.
Subtract c from both sides of the
equation. 
ax^{2} + bx + c = 0 ax^{2}
+ bx = c 
Step 2 If the coefficient of x^{2} is not 1,
divide both sides of the equation
by the coefficient of x^{2}.
The coefficient of x^{2} 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 xterm 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 4a^{2}. 

Combine like terms on the
right side. In the numerator,
write the b^{2}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 4a^{2} = 2a.
If a < 0, then 4a^{2 }= 2a.
So,
The result is called the quadratic formula.
Formula The Quadratic Formula
The solutions of the quadratic equation ax^{2} + bx + c = 0 are given
by the quadratic formula:
Here, a, b, and c are real numbers and a ≠
0.
