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
z x2 - 10x + 25
z x2 + 12x + 36
z
Try to factor these (they’re easy).
Perfect Square Trinomials
z x2
+ 6x + 9
z x2 - 10x + 25
z 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
z
z
z
z
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.
z x2 + 20x + ___
z x2 - 4x + ___
z x2 + 5x + ___
z
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:
2 x − 7 x + 12 = 0
Step 1: Move the constant
to the right side of the
equation.
2 x − 7 x = −12
2
2
Solving Quadratic Equations
by Completing the Square
2 x − 7 x + , =-12 +,
Step 2: Find the term
2x 7x
12
that completes the square
−
+ , =− + ,
on the left side of the
2
2
2
2
2
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.
7
x − x +,= −6 +,
2
2
1 7
7
• ( ) = then square it,
2 2
4
7
49
49
x − x+
= −6 +
2
16
16
2
2
49
⎛7⎞
⎜ ⎟ =
⎝ 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
⎝