Download Quadratic Equations: Completing the Square

Quadratic Equations: Completing the Square
Definition Completing the square is an algebraic technique that
transforms any quadratic equation into the form Q2 = k.
Remark 1 The factoring method and the square root property each allow us to
solve SOME quadratic equations. Mastering the technique of completing the square will allow us to solve ALL quadratic equations.
The key to understanding completing the square is to first understand
perfect square trinomials!
Definition A perfect square trinomial is a three-term polynomial
that can be factored into a binomial squared.
ex.
−→
Perfect
square trinomial:
x2 + 6x + 9
x2 − 10x + 25
Factored form:
−→
−→
(x + 3)(x + 3) = (x + 3)2
(x − 5)(x − 5) = (x − 5)2
Remark 2 Any binomial of the form x2 + Bx can be transformed into a
perfect square trinomial by adding the appropriate number!
ex.
−→ Fill in the blank with the appropriate number to transform the binomial
of the form x2 + Bx into a perfect square trinomial.
x2 + 12x +
= (x + 6)2
To see which number will balance the equation, let’s expand the right
side.
(x + 6)2 = (x + 6)(x + 6) = x2 + 12x + 36
Comparing this to the left side we see that the blank must be filled in
with 36 .
Remark 3 There is an important relationship between a perfect square trinomial
and its factored form that should be recognized here:
x2 + 12 x + 36 = (x + 6 )2
↗
↖
In this trinomial
B = 12
In the factored form, the number being
added to x is half of B, that is 6 = 12 B.
Remark 4 Let’s further explore the idea of creating perfect square trinomials.
ex.
−→ Fill in the blanks with two different numbers to transform the binomial of the form x2 + Bx into a perfect square trinomial.
x2 − 18x +
)2
= (x +
Notice that B = −18. We can fill in the blue blank with a simple
calculation (reconsider remark 3):
1
2B
= 21 (−18) = −9
Now we just have to fill in the pink blank:
x2 − 18x +
= (x − 9)2
Expanding the right side we see that the pink blank must be 81 .
Remark 5 We can now expand on the relationship between a binomial of the form
x2 + Bx and the corresponding perfect square trinomial.
x2 −18 x +
↑
B = −18
= (x −9 )2 = x2 − 18x + 81
↑
(
)2 ↑
1
B
2
= −9
1
B
2
= 81
Technique To complete the square on an equation of the form x2 + Bx = k,
( )2
add 12 B to both sides of the equation.
Main Idea To solve a quadratic equation by completing the square:
1 Manipulate the equation so that one side looks like x2 + Bx.
→ The squared variable must have a coefficient of 1!
( )2
2 Add 12 B to both sides.
(
)2
3 Factor the perfect square trinomial into x + 12 B .
4 Solve the equation using the square root property.
Example 1 Solve each equation by completing the square.
a x2 + 10x − 4 = 0
b x2 = 18x − 69
c (x − 10)(x + 2) = −30
d
2y 2 + 5y + 1 = 0