Download Solving Quadratic Equations by Completing the Square

Int. Alg. Notes
Section 8.1
Page 1 of 5
Sections 8.1: Solving Quadratic Equations by Completing the Square
Big Idea: Quadratic equations that can’t be factored can still be solved by converting the quadratic equation to
the square of a binomial equals a constant, and then taking the square root of both sides.
Big Skill: You should be able to solve any quadratic equation by completing the square.
The Square Root Property
If x 2  p then x  p or x   p .
To Solve a Quadratic Equation Containing Only a Square and Constant Term:

Isolate the square term.

Use the square root property.
Comparison Example:
Solve x 2  16  0 using factoring.
x 2  16  0
 x  4  x  4   0
x  4  0 OR
x40
x4
OR
x  4
Solve x 2  16  0 using the square root property.
x 2  16  0
x 2  16
x  16 OR
x   16
x4
x  4
OR
Note: Instead of writing
x  16 OR x   16 ,
mathematicians use the shorthand
“plus-minus” notation of
x   16
.
x  4
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.1
Page 2 of 5
Another Example: Note that the square term can be the square of a quantity.
 y  3  100  0
2
 y  3  100
2
y  3  100
OR
y  3   100
y  3  10
OR
y  3  10
y7
OR
y  13
Here is how to write the solution using “plus-minus” notation:
 y  3
2
 100
 y  3
2
  100
y  3  10
y  3  10
y  3  10 OR
y  3  10
y7
y  13
OR
Practice: Solve the following quadratic equations:
1. z 2  24  0
2. 7r 2  112
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.1
Page 3 of 5
3. z 2  16  4
4.
 x  3
5.
 a  2
2
2
 16
 12  0
The last complication in solving quadratic equations occurs when the quadratic equation has a linear term and is
prime (i.e., it can’t be factored), like x 2  2 x  1  0 .
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.1
Page 4 of 5
In cases like this, we manipulate the equation so that it becomes the square of a binomial plus a constant, like in
practice problems #4 and #5. This manipulation involves taking the first two terms, and finding out what we
have to add to them to make a perfect square trinomial, which can be replaced with the square of a binomial.
So, for x 2  2 x  1  0 , the first two terms are identical to the first two terms of the perfect square trinomial
2
x 2  2 x  1 , which comes from the square of the binomial x + 1:  x  1  x 2  2 x  1 . So, here is what we do:
x2  2x 1  0
x2  2x  1
x2  2x  1  1  1
 x  1
2
2
 x  1
2
 2
x 1   2
x  1  2
x  1  2
OR
x  1  2
x  0.414
OR
x  2.414
To Solve a Quadratic Equation by Completing the Square :
(i.e, writing a quadratic trinomial as a perfect square trinomial plus a constant)
 Get the constant term on the right hand side of the equation.
i.e., if x 2  bx  c  0 , then write the equation as x 2  bx  c
 Make sure the coefficient of the square term is 1.
 Identify the coefficient of the linear term; multiply it by ½ and square the result.
2

1 
i.e., Find the number b in x  bx  c and compute  b 
2 
Add that number to both sides of the equation.

1 
1 
i.e., x  bx   b   c   b 
2 
2 
Write the resulting perfect square trinomial as the square of the binomial .

1 

1 
i.e.,  x  b   c   b 
2 

2 
Use the square root property to solve the equation.
2
2
2
2
2
2
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 8.1
Page 5 of 5
Practice: Solve the following quadratic equations:
6. n 2  10n  6  0
7. x 2  16 x  7  0
8. z 2  7 z  1  0
9. 3 x 2  6 x  5  0
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.