Ma 15200 Lesson 14 Section 1.5 Part 1 - Quadratic Equations Worksheet - Purdue University Page 3
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6III
Solving by Completing the Square
. Half
2
A perfect square trinomial with a leading coefficient of 1 has the form
x
bx c
2
1
of the middle coefficient squared equals the last term.
If the leading
b
c
2
x
2
coefficient is a 1, every binomial of the form
bx
can be made into a perfect square
2
b
1
trinomial by adding
. If this value is added to both sides of an equation of the
2
2
form
x
bx
c
, then the square root property can be used to solve the equation. This
process is called the completing the square process.
Here is an example:
2
x
10
x
24
0
2
x
10
x
24
2
2
1
1
2
x
10
x
10
24
10
2
2
2
x
10
x
25
24
25
2
(
x
) 5
1
x
5
1
and
x
5
1
x
5
1
x
5
1
x
4
x
6
*There is a good visual picture of completing the square on page 141 of the textbook.
Ex 3: What number should be added to each binomial so that it becomes a perfect
square trinomial?
2
a
)
x
14
x
2
b
)
r
22
r
2
c
)
n
3
n
Any quadratic equation can be solved using completing the square if each term is divided
by the leading coefficient. However, it is recommended you only use this method when
the leading coefficient is already a 1.
Here are the steps.
1. Use this procedure only on equations where the leading coefficient is 1.
2
2. Move the constant to the other side so the equation is in the form
x
bx
c
.
3. Complete the square by taking half of the coefficient of x, squaring it, and adding
to both sides of the equation.
3
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