Completing the Square and The Square Root Method
The Square Root Property
If we have
x2 = k
then we can take the square root of both sides to solve for x.
The Square Root Property
For any positive number k,
if
x2
= k
then
x =
or x = -
|
Example
Solve
x2 - 6 = 0
Solution
First add 6 to both sides
x2 = 6
Next use the square root property
x =
or x = -
Example
Solve
(x - 3)2 + 5 = 12
Solution
(x - 3)2 =
7
Subtract
5 from both sides
x
- 3 =
or x - 3 = -
Use the square root property
x = 3 +
or x = 3 -
Add 3 to both sides
Caution: The square root
property cannot be directly applied in a quadratic that has a middle term such as
x2 + 5x - 2
Completing The Square
We have seen that the square root property only worked when the middle term was zero. For
example if
3(x - 1)2 -
3 = 0
then we can use the square root property. A quadratic is said to be in standard
form if it has the form
a(x - h)2 + k Standard
Form of a Quadratic
If we are given a quadratic in the form
ax2
+ bx + c
We would like to put the quadratic into standard form so that we can use the
square root property. We call the process of putting a quadratic into
standard form Completing the Square.
Below is a step by step process of completing the square.
Example
Complete the Square
2x2 -
8x + 2 = 0
Solution
-
Factor the leading coefficient from the first two terms:
2(x2 -
4x) + 2
-
Calculate b/2:
-4
= -2
b is the coefficient in
front of the "x" term.
2
-
Square the solution above:
22
= 4
-
Add and subtract answer from part three (the magic
number) inside parentheses:
2(x2 -
4x + 4 - 4) + 2
-
Regroup:
2[(x2 - 4x + 4 ) - 4] + 2
-
Factor the inner parentheses using part two as a hint:
2[(x
- 2)2 - 4] + 2
-
Multiply out the outer constant:
2(x
- 2)2 - 8 + 2
-
Combine the last two constants:
2(x - 2)2
- 6
-
Breath a sigh of relief.
Example
Complete the square
3x2
+ 5x + 1
Solution
- 3(x2 + 5/3 x) + 1
Pulling
a
3 out of a five is the same as dividing
5 by
3
- b/2 = 5/6
- (5/6)2 = 25/36
Square b/2
- 3(x2 + 5/3 x + 25/36 -
25/36) + 1 Add and subtract the magic number (b/2)2
- 3[(x2 + 5/3 x + 25/36) - 25/36] + 1
Regroup
- 3[(x + 5/6)2 - 25/36] + 1
Factor the first three terms
- 3(x + 5/6)2 - 25/12 + 1
Multiply the 3 through
- 3(x + 5/6)2 -
13/12
Note: -25/12 + 1 = -25/12 +12/12 = -13/12
Exercises:
Complete the square
- 3x2 - 12x + 6
- 2x2 - 2x + 4
- 4x2 + 4x - 3
Practice
completing the square
Completing the Square to Solve a Quadratic Equation
Example
Solve
x2 +
2x - 5 = 0
Solution
We see that there is a middle term, 2x,
so the square root property will not work. We first complete the square. We
have
(b/2)2
= 1
x2 +
2x + 1 - 1 - 5
= 0
Adding
and subtracting 1
(x + 1)2
- 6 = 0
Factoring
the first three terms
Now we can use the square root property
(x + 1)2
= 6
Adding 6 to both sides
x + 1
=
or x + 1 = -
Taking the square root of both sides
x =
-1 +
or x = -1 -
Subtracting 1 from both sides
Example
Solve
x2
+ 6x + 13 = 0
Solution
We see that there is a middle
term, 6x, so the square root property will not
work. We first complete the square. We have
(b/2)2
= 9
x2
+ 6x + 9 - 9 +
13 = 0 Adding
and subtracting the magic number 9
(x + 3)2
+ 4 = 0 Factoring
the first three terms
Now we can use the square root property
(x + 3)2
= -4
Subtract 4 from both sides
x + 3
=
or x + 3 = -
Taking the square root of
both sides
x =
-3 +
or x = -3 -
Subtracting 1 from both sides Notice
that is
not a real number but we can still write the imaginary solutions since
=
2i The
final solutions are
x = -3 + 2i or x
= -3 - 2i
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and Linear Inequalities Page
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