Completing the Square
for Conic Sections
The Aim of Completing the Square
… is to write a quadratic function as a perfect square. Here are
some examples of perfect squares!
x2 + 6x + 9
 x2 - 10x + 25
 x2 + 12x + 36

Try to factor these (they’re easy).
Perfect Square Trinomials
 x2
+ 6x + 9
 x2 - 10x + 25
 x2 + 12x + 36
=(x+3)2
=(x-5)2
=(x+6)2
Can you see a numerical connection between …
6 and 9 using 3
-10 and 25 using -5
12 and 36 using 6
The Perfect
Square Connection
For a perfect square, the following relationships will
always be true …
x2 + 6x + 9
Half of these values
squared
x2 - 10x + 25
… are these values
The Perfect
Square Connection




In the following perfect square trinomial,
the constant term is missing. Can you
predict what it might be?
X2 + 14x + ____
Find the constant term by squaring half
the coefficient of the linear term.
(14/2)2
X2 + 14x + 49
Perfect Square Trinomials
Create perfect
square trinomials.
 x2 + 20x + ___
 x2 - 4x + ___
 x2 + 5x + ___

100
4
25/4
Solving Quadratic Equations
by Completing the Square
Solve the following
equation by
completing the
square:
Step 1: Move the
constant term (i.e.
the number) to
right side of the
equation
x  8 x  20  0
2
x  8 x  20
2
Solving Quadratic Equations
by Completing the Square
Step 2:
Find the term
that completes the square
on the left side of the
equation. Add that term
to both sides.
x  8x 
2
=20 +
1
 (8)  4 then square it, 42  16
2
x  8 x  16  20 16
2
Solving Quadratic Equations
by Completing the Square
Step 3:
Factor
the perfect
square trinomial
on the left side
of the equation.
Simplify the
right side of the
equation.
x  8 x  16  20 16
2
( x  4)( x  4)  36
( x  4)  36
2
For chapter 10 material, we can stop here. But solving is a
simple process from here …
Solving Quadratic Equations
by Completing the Square
Step 5: Set
up the two
possibilities
and solve
x  4  6
x  4  6 and x  4  6
x  10 and x=2
Completing the Square-Example #2
Solve the following
equation by completing
the square:
Step 1: Move the constant
to the right side of the
equation.
2 x  7 x  12  0
2
2 x  7 x  12
2
Solving Quadratic Equations
by Completing the Square
Step 2:
Find the term
that completes the square
on the left side of the
equation. Add that term
to both sides.
The quadratic coefficient
must be equal to 1 before
you complete the square, so
you must divide all terms
by the quadratic
coefficient first.
2 x2  7 x 
=-12 +
2 x2 7 x
12


 
2
2
2
2 7
x  x   6 
2
1 7 7
 ( )  then square it,
2 2 4
7
49
49
x  x
 6 
2
16
16
2
2
 7  49
  
 4  16
Solving Quadratic Equations
by Completing the Square
Step 3:
Factor
the perfect
square trinomial
on the left side
of the equation.
Simplify the
right side of the
equation.
Use calculator to do this!
7
49
49
x  x
 6 
2
16
16
2
2
7
96 49

x   
4
16 16

2
7
47

x  
4
16

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The Aim of Completing the Square