Use Square Roots to Solve
Quadratic Equations
Andrew Gloag
Eve Rawley
Anne Gloag
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Printed: May 5, 2015
AUTHORS
Andrew Gloag
Eve Rawley
Anne Gloag
www.ck12.org
C HAPTER
Chapter 1. Use Square Roots to Solve Quadratic Equations
1
Use Square Roots to Solve
Quadratic Equations
Here you’ll learn how to solve quadratic equations in which finding the solutions involves square roots.
What if you had a quadratic equation like 4x2 − 9 = 0 in which both terms were perfect squares? How could you
solve such an equation? After completing this Concept, you’ll be able to solve quadratic equations like this one that
involve perfect squares.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/133106
CK-12 Foundation: 1004S Solving Quadratic Equations Using Square Roots
Guidance
So far you know how to solve quadratic equations by factoring. However, this method works only if a quadratic
polynomial can be factored. In the real world, most quadratics can’t be factored, so now we’ll start to learn other
methods we can use to solve them. In this Concept, we’ll examine equations in which we can take the square root
of both sides of the equation in order to arrive at the result.
Solve Quadratic Equations Involving Perfect Squares
Let’s first examine quadratic equations of the type
x2 − c = 0
We can solve this equation by isolating the x2 term: x2 = c
Once the x2 term is isolated we can take the square root of both sides of the equation. Remember that when we take
the square root we get two answers: the positive square root and the negative square root:
x=
√
c
and
√
x=− c
√
Often this is written as x = ± c.
Example A
Solve the following quadratic equations:
1
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a) x2 − 4 = 0
b) x2 − 25 = 0
Solution
a) x2 − 4 = 0
Isolate the x2 : x2 = 4
Take the square root of both sides: x =
√
√
4 and x = − 4
The solutions are x = 2 and x = −2.
b) x2 − 25 = 0
Isolate the x2 : x2 = 25
Take the square root of both sides: x =
√
√
25 and x = − 25
The solutions are x = 5 and x = −5.
We can also find the solution using the square root when the x2 term is multiplied by a constant—in other words,
when the equation takes the form
ax2 − c = 0
We just have to isolate the x2 :
ax2 = b
b
x2 =
a
Then we can take the square root of both sides of the equation:
r
x=
r
Often this is written as: x = ±
b
a
r
and
b
.
a
Example B
Solve the following quadratic equations.
a) 9x2 − 16 = 0
b) 81x2 − 1 = 0
Solution
a) 9x2 − 16 = 0
Isolate the x2 :
9x2 = 16
16
x2 =
9
2
x=−
b
a
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Chapter 1. Use Square Roots to Solve Quadratic Equations
r
Take the square root of both sides: x =
Answer: x =
4
3
16
and x = −
9
r
16
9
and x = − 43
b) 81x2 − 1 = 0
Isolate the x2 :
81x2 = 1
1
x2 =
81
r
r
1
1
and x = −
Take the square root of both sides: x =
81
81
Answer: x =
1
9
and x = − 19
As you’ve seen previously, some quadratic equations have no real solutions.
Example C
Solve the following quadratic equations.
a) x2 + 1 = 0
b) 4x2 + 9 = 0
Solution
a) x2 + 1 = 0
Isolate the x2 : x2 = −1
Take the square root of both sides: x =
√
√
−1 and x = − −1
Square roots of negative numbers do not give real number results, so there are no real solutions to this equation.
b) 4x2 + 9 = 0
Isolate the x2 :
4x2 = −9
9
x2 = −
4
r
r
9
9
Take the square root of both sides: x = − and x = − −
4
4
There are no real solutions.
We can also use the square root function in some quadratic equations where both sides of an equation are perfect
squares. This is true if an equation is of this form:
(x − 2)2 = 9
Both sides of the equation are perfect squares. We take the square root of both sides and end up with two equations:
x − 2 = 3 and x − 2 = −3.
3
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Solving both equations gives us x = 5 and x = −1.
Example D
Solve the following quadratic equations.
a) (x − 1)2 = 4
b) (x + 3)2 = 1
Solution
a) (x − 1)2 = 4
Take the square root of both sides :
x − 1 = 2 and x − 1 = −2
x = 3 and x = −1
Solve each equation :
Answer: x = 3 and x = −1
b) (x + 3)2 = 1
Take the square root of both sides :
x + 3 = 1 and x + 3 = −1
Solve each equation :
x = −2 and x = −4
Answer: x = −2 and x = −4
It might be necessary to factor the right-hand side of the equation as a perfect square before applying the method
outlined above.
Watch this video for help with the Examples above.
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CK-12 Foundation: 1004 Solving Quadratic Equations Using Square Roots
Vocabulary
• The solutions of a quadratic equation are often called the roots or zeros.
Guided Practice
Solve the following quadratic equations.
a) x2 + 8x + 16 = 25
b) 4x2 − 40x + 25 = 9
4
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Chapter 1. Use Square Roots to Solve Quadratic Equations
Solution
a) x2 + 8x + 16 = 25
Factor the right-hand-side :
x2 + 8x + 16 = (x + 4)2
Take the square root of both sides :
x + 4 = 5 and x + 4 = −5
Solve each equation :
x = 1 and x = −9
so (x + 4)2 = 25
Answer: x = 1 and x = −9
b) 4x2 − 20x + 25 = 9
Factor the right-hand-side :
4x2 − 20x + 25 = (2x − 5)2
Take the square root of both sides :
2x − 5 = 3 and 2x − 5 = −3
Solve each equation :
2x = 8 and 2x = 2
so (2x − 5)2 = 9
Answer: x = 4 and x = 1
Explore More
Solve the following quadratic equations.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
x2 − 1 = 0
x2 − 100 = 0
x2 + 16 = 0
9x2 − 1 = 0
4x2 − 49 = 0
64x2 − 9 = 0
x2 − 81 = 0
25x2 − 36 = 0
x2 + 9 = 0
x2 − 16 = 0
x2 − 36 = 0
16x2 − 49 = 0
(x − 2)2 = 1
(x + 5)2 = 16
(2x − 1)2 − 4 = 0
(3x + 4)2 = 9
(x − 3)2 + 25 = 0
x2 − 10x + 25 = 9
x2 + 18x + 81 = 1
4x2 − 12x + 9 = 16
2(x + 3)2 = 8
5