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Quadratic Equations
Completing the Square is also useful for solving the quadratic equations.
Example 1 Solve x2 6x + 7 = 0 by completing the square.
Solution) Moving the constant 7 from left to right,
x2
6x =
7
Adding the secret number to both sides (why 9?)
x2
6x +9 =
7 +9
Completing the square,
3)2 = 2
(x
Taking the square root,
x
Moving the constant
3=
p
2
3 from left to right,
p
x=3
2
Note: Last time you completed the square:
y = 2 x2
6x + 10 = 2 x2
6x +9
9
+ 10 = 2 (x
3)2
8
In that case, you worked with equivalent expressions, so inserted +9 9 to keep the equality. On the
contrary, you are working with equivalent equations when solving x2 6x + 7 = 0. So you add 9 to both
sides to keep the equality. Do not put the equal sign at the beginning. For example, writing such as
= x2
6x + 9 =
7+9
is WRONG.
Example 2
Solve x2
6x + 7 = 0 by the quadratic formula.
The solutions of ax2 + bx + c = 0 are
x=
Solution) Identify a = 1; b =
b
p
b2
2a
4ac
6; c = 7. Hence
q
p
( 6)
( 6)2 4 (1) (7)
6
8
x=
=
2 (1)
2
p
p
6 2 2
=
=3
2
2
Note: The 2nd method looks much easier than the 1st, but how do we get the quadratic formula? By
completing the square of ax2 + bx + c = 0!! The derivation of the general formula is a little complicated,
so ok to skip. But you must be able to solve the quadratic equations by both methods for the speci…c
examples.
Example 3 Solve x2 6x + 8 = 0.
Solution) Factoring the LHS, we get
(x
2) (x
4) = 0
Since the product is zero,
x
2 = 0 or x
4=0
Solving each, we get
x = 2; 4
Note: You could have used completing the square, or quadratic formula for this example, but factoring
is much easier. On the other hand, you cannot use factoring for Example 1 and 2.
Example 4 Solve x2
Solution) Factoring,
6x + 9 = 0.
3)2 = 0
(x
Hence,
x
3=0
So,
x=3
Note: Quadratic equations usually have two solutions (called roots). Although there is only one root in
this example, we often view that two roots happen to coincide. So this type of root is called a double root.
Example 5 Solve x2 6x + 10 = 0 by two methods.
Solution) Factoring is not available, but other two methods work.
(1) Completing the Square
x2 6x + 13 = 0
x2
6x
=
x2
(x
x
x
x
6x + 9
3)2
3
3
= 13 + 9
= 4p
= p 4
=
4i
= 3 2i
13
(2) Quadratic Formula
x2 6x + 13
p = 02
( 6)
( 6) 4(1)(13)
x=
2(1)
p
= 6 p2 16
= 6 2 16i
= 6 24i
= 3 2i
p
Note: No real numbers satisfy x2 = 1. So we de…ne a new number i =
1 (square root of 1)
2
satisfying this equation. Hence ip
= 1. p
Using i, we can solve any quadratic equation. When a > 0,
a=
ai.
x2 = a has two solutions x =