7.3 Part 1 Simplifying Radicals and Solving Quadratic Equations by Taking Square Roots
Part 1: Simplifying Square Roots
List as many perfect squares as you can in the box below:
Now find the value of their square roots.
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Not every number has a perfect square root. Estimate the following radicals without a calculator.
13  ______
3  ______
29  ______
91  ______
55  ______
117  ______
With fractions, we always write our answers in simplest form (reduce, no decimals!!!!!!!)
24
18
With radicals, we always write our answer in simplest form as well (reduce by perfect squares, no decimals)
Method 1: Factor Tree
Method 2: Perfect Squares
Example:
12
12
Example:
27
27
Example:
3 40
3 40
Your Turn!
Simplify each square root.
1.
18
2.
4.
8 84
5.
54
12 300
3.
60
6.
7 425
Part 2: Rationalizing the Denominator
When simplifying radicals, we may never have any radicals in the denominator, so we perform a process called
RATIONALIZING THE DENOMINATOR, meaning we want the bottom of the fraction to end up with just the number, no
radical.
Follow the steps to rationalize the denominator.
Example1:
3
5
3
5

5 5
3 5
25
3 5
5
Example 2:
2
8
2
8

8 8
2 8
64
There is as radical in the denominator.
Multiply the numerator and denominator by the radical that we want to get rid of.
Simplify.
Simplify. 5 does not go into 3, so the fraction cannot be reduced any further.
There is a radical in the denominator.
Multiply the numerator and denominator by the radical that we want to get rid of.
Simplify.
2 8 22 2 4 2
2



8
8
8
2
Simplify.
3 2
5 7
Example 3:
There is a radical in the denominator.
3 2 7

5 7 7
Multiply the numerator and denominator by the radical that we want to get rid of.
3 14
5 49
Simplify
3 14 3 14

57
35
Square root of 14 cannot be simplify and 35 and 3 cannot be simplified, so we’re done.
Your Turn!
Rationalize the denominator. Hint: Simplify radicals, divide radicals, and take perfect squares before rationalizing
denominators to make your work easier.
1.
4.
12
10
1
√2
8)
2.
5.
2
5√7
13
32
2
√4
9.
3.
6)
3
7
√3
√5
9 20
12 48
7)
10.
5
12
2
√7 √4
Part 3: Solving Quadratic Equations by Taking Square Roots
Solve x2 = 18
Solve (x + 4)2 – 3 = 17
Solve 6x2 – 240 = 0
Solve 3x2 + 21 = 5
Solve 3x2 – 7 = 4
Your Turn to Practice!
Solve each equation. Leave your answer in simplest form and round your answer
to the nearest hundredth.
1. (3x + 1)2 = 15
2. (x – 3)2 = 12
3. (2x – 5)2 = 24
4. (7x – 5)2 – 75 = 0
5. 3x2 – 27 = 0
6. (x – 2)2 – 25 = 0
7. (x + 1)2 -
9.
9
4
1
( x 5) 2  7
3
=0
8. (x – 3)2 + 18 = 0
10. 5(2x + 1)2 – 6 = 6
Name: _________________________
Date: ________________________
Homework 7.3 part 1 – Square Roots and Equations
Simplify each square root and rationalize the denominator.
1)
4√10
√2
2)
3
3)
√5
2
4)
√90
5) √50
6) √48
7) √3245
8) √18
9) 4√8
10) 3√16
17
√2∙√3
11) 5√125
Solve each equation for x. Leave your answer in simplest form AND round your
answer to the nearest hundredth.
12)
2(3x – 8)2 – 8 = 0
13)
5x2 = 80
14)
5x2 – 49 = 0
15)
3x2 – 24 = 0
16)
2x2 - 64 = 0
17)
4x2 = 27
Name: ______________________________
Exit Ticket 7.3 part 1
1)
Solve the equation by factoring
9x2 – 25 = 0
2) Solve the equation by square root
9x2 – 25 = 0
3) Which method do you prefer?
4) How do you know when to use either method?
Name: ______________________________
Exit Ticket 7.3 part 1
1)
Solve the equation by factoring
9x2 – 25 = 0
2) Solve the equation by square root
9x2 – 25 = 0
3) Which method do you prefer?
4) How do you know when to use either method?