22 1. REVIEW OF ALGEBRA
1.3.2 COMPLETING THE SQUARE
We can’t always find the roots of a quadratic by factoring. Another way to find the roots is to
transform the quadratic from the form:
ax
2
C bx C c D 0
into the form:
.x r/
2
D s
en, we can find the roots by taking the square root of both sides and doing some algebra.
is is a technique called completing the square. Completing the square consists of forcing a
complete square in the form .x r/
2
and then putting the leftovers on the other side of the
equation. Notice that:
.x a/
2
D x
2
2ax C a
2
To complete the square we follow these steps:
1. From the equation above, we see that the constant term must be the square of half the
number in front of x. We can force this by adding and subtracting the needed constant.
2. Step 1 transforms the equation into a perfect square plus a constant. Take the constant to
the other side.
3. Solve by using the square root and normal algebra. Remember the ˙ on the square root—
we will get two answers.
Example 1.47 Solve x
2
C 2x 1 D 0 by completing the square.
Solution:
x
2
C 2x 1 D 0 is is the original equation
x
2
C 2x C .1 1/ 1 D 0 Add and subtract
1
2
2
2
x
2
C 2x C 1 2 D 0 Regroup to get perfect square and constant
.x C 1/
2
2 D 0 Make the perfect square explicit by factoring
.x
C
1/
2
D
2
Move 2 to the other side
x C 1 D ˙
p
2
Take the square root
x D 1 ˙
p
2 Move 1 to the other side
And we see the solutions to the quadratic equation are x D 1
p
2 and x D 1 C
p
2. is
also means
x
2
C 2x 1 D .x C 1 C
p
2/.x C 1
p
2/:
1.3. QUADRATIC EQUATIONS 23
is factorization is not easy to see—which is why we need tools like completing the square.
˙
ere is another thing that completing the square can do.
Example 1.48 Use completing the square to show x
2
4x C 7 D 0 does not have any solu-
tions.
Solution:
x
2
4x C 7 D 0 is is the original equation
x
2
4x C .4 4/ C 7 D 0 Add and subtract
1
2
4
2
x
2
4x C 4 C 3 D 0 Regroup to get perfect square and constant
.x 2/
2
C 3 D 0 Make the perfect square explicit by factoring
.x 2/
2
D 3 Move 3 to the other side
x
2
D ˙
p
3 Take the square root. D’oh! Negative!!
˙
Example 1.48 generated a root that was a square root of a negative. In fact, the right-hand side
of the equation, just before we take the square root, is exactly the discriminant D D b
2
4ac.
Knowledge Box 1.9 told us that when the discriminant is negative, there are no roots.
Another nice thing about completing the square is that it gives us the geometrically important
parts of the quadratic. Quadratic equations with a positive squared term, including all the ones
we’ve seen so far in this section, graph as parabolas opening upward. ose with a negative
squared term open downward. ese show up in falling object equations. e root of the perfect
square that shows up when we are completing the square is the point where the quadratic turns
around and heads in the other direction. If we have a downward-opening quadratic, this tells us
where the highest point on the quadratic is. e next example is a physics-like example, so we
are going to use time t as the independent variable instead of x.
Example 1.49 Suppose that the height in meters of a thrown ball after t seconds is given by:
h D 6 C 4t t
2
m
Complete the square to find when the ball is highest and at what time that greatest height occurs.
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