Solving a quadratic by Completing the Square
To solve a quadratic equation by completing the square:
1. Write the equation in the form x2 + bx = c
2. Complete the square by adding (b/2)2 to both sides.
3. Write the completed square in factored form .
4. Use the square root principle to eliminate the
5. Isolate the variable.
6. Simplify as needed .
Solve each of the following by completing the square:
1. x2 – 12x – 5 = 0
2. 9x2 + 18 x = -1
Instead of completing the square on each problem, let’s
complete the square on the general quadratic equation and derive a formula.
Let’s derive the Quadratic Formula
ax2 + bx + c = 0
ax2 + bx + c = 0 if and only if
Now let’s use it to solve:
1. x2 + 3x – 8 = 0
a = _____ b = _____ c = _____
2. x2 – 9x = 41
First you must write your equation in the correct form.
a = ___ b =___ c = ___
We have now studied several methods for solving a
quadratic equation. The following table summarizes the methods.
When the Method is Best
If the quad. can be factored easily
2. Square Root
If quad can easily be written in ax2 = c or (ax
+ b)2 = c
3. Completing the square
4. Quadratic formula
When factoring is not easy.
The expression b2 – 4ac is called the discriminant.
Given a quadratic equation in the form ax2 + bx + c = 0, where a ≠ 0, to
determine the number and type of solution the quadratic has we will evaluate the
b2 –4ac < 0, then the equation has two non real
b2 –4ac = 0, then the equation has one real solution
b2 –4ac > 0, then the equation has two real solutions.
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