1. Solve Quadratic Equations by Finding Square Roots Lesson 1.5
Honors Algebra 2
Solve Quadratic Equations by Finding Square Roots
Lesson 1.5

2. Goals Goal Rubric Simplify radicals. Rationalize the denominator.
Solve Quadratic Equations by finding square roots.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Square root Radical Radicand Rationalizing the denominator
Conjugates

4. These numbers are called the
Perfect Squares
The terms of the following sequence:
1, 4, 9, 16, 25, 36, 49, 64, 81…
12,22,32,42, 52 , 62 , 72 , 82 , 92…
These numbers are called the
Perfect Squares.

5. Square Roots The number r is a square root of s, if r2 = s.
This is usually written 𝑠 =𝑟.
Any positive number has two real square roots, one positive and one negative, 𝑠 and − 𝑠 = 2 and -2, since 22 = 4 and (-2)2 = 4
The positive square root is called the principal square root.
The expression 𝑠 is called a radical.
The symbol is a radical sign. The number or expression under the radical symbol is called the radicand.

6. Properties of Square Roots
Square roots have special properties that help you simplify, multiply, and divide them.

7. Simplifying Square Root
The properties of square roots allow us to simplify radical expressions.
A radical expression is in simplest form when:
The radicand has no perfect-square factor other than 1.
and
There’s no radical in the denominator.

8. EXAMPLE 1
Use properties of square roots
Simplify the expression. a. 80 5 16 = 5
= 4 b. 6 21
126 = 9 14 =
= 3 14 c. 4 81 = 4 81 =
2 9 d. 7 16 = 7 16 = 4 7

9. GUIDED PRACTICE Your Turn: for Example 1 27 3 98 2
Simplify the expression. 27 3
ANSWER 98 2 7
ANSWER

10. GUIDED PRACTICE Your Turn: for Example 1 10 15 6 5 8 28 14
Simplify the expression. 10 15
ANSWER 6 5 8 28
ANSWER 14 4

11. GUIDED PRACTICE Your Turn: for Example 1 9 64 3 8 15 4 2 15
Simplify the expression. 9 64
3 8
ANSWER 15 4 2 15
ANSWER

12. GUIDED PRACTICE Your Turn: for Example 1 11 25 5 11 36 49 7 6
Simplify the expression. 11 25 5 11
ANSWER 36 49 7 6
ANSWER

13. Rationalizing the Denominator
If a fraction has a denominator that is a square root, you simplify it by rationalizing the denominator.
To do this, multiply both the numerator and denominator by a number that produces a perfect square under the radical sign in the denominator. That simplifies to a rational number.

14. Rationalizing the Denominator
For a fraction with a denominator of the 𝑏 or 𝑎± 𝑏 , The table below shows how to eliminate the radical from the denominator.
The expressions 𝑎+ 𝑏 and 𝑎− 𝑏 are conjugates. The product of conjugates is always a rational number.

15. EXAMPLE 2
Rationalize denominators of fractions. 5 2 3
7 + 2
Simplify
(a)
and
(b)
SOLUTION
(a) 5 2 = 5 2 = 5 2 2 10 =

16. EXAMPLE 2
Rationalize denominators of fractions.
SOLUTION = 3
7 + 2
7 –
(b) 3
7 + 2 =
21 – 3 2
49 – – 2 =
21 – 3 2 47

17. GUIDED PRACTICE Your Turn: for Example 2 6 5 9. 5 30 9 8 10. 2 4 3
Simplify the expression. 6 5 9. 5 30
ANSWER 9 8
10. 2 4 3
ANSWER

18. GUIDED PRACTICE Your Turn: for Example 2 17 12 11. 51 6 19 21 399 21
Simplify the expression. 17 12
11. 51 6
ANSWER 19 21
399 21
ANSWER

19. Your Turn: for Example 2 – 6 7 – 5 – 21 – 3 5 22 2 4 + 11 8 – 2 11 5
Simplify the expression.
– 6
7 – 5
– 21 – 3 5 22
ANSWER 2
4 + 11
8 – 2 11 5
ANSWER

20. Your Turn: for Example 2 – 1 9 + 7 – 9 + 7 74 4 8 – 3 32 + 4 3 61
Simplify the expression.
– 1
9 + 7
– 9 + 7 74
ANSWER 4
8 – 3
32 + 4 3 61
ANSWER

21. Solving Quadratic Equations by Finding Square Roots
Some types of quadratic equations can be solved by finding square roots.
Quadratic equations of the form x2 = s (the equation contains a perfect square term and has no linear term), where x is an algebraic expression and s is a nonzero real number, can be solved by finding Square Roots.
If x is an algebraic expression and s is a nonzero real number, then x2 = s has exactly two solutions:
𝑥= 𝑠 or 𝑥=− 𝑠
Equivalently, if x2 = s, then 𝑥=± 𝑠 .

22. Solving Quadratic Equations by Finding Square Roots
Procedure
Step 1: Isolate the expression containing the square term.
Step 2: Take the square root of both sides of the equation. Don’t forget the  symbol.
Step 3: If necessary, isolate the variable and solve.
Step 4: Check. Verify your solutions.

23. EXAMPLE 3 Solve 3x2 + 5 = 41. 3x2 + 5 = 41 3x2 = 36 x2 = 12 x = + 12
Solve a quadratic equation
Solve 3x2 + 5 = 41.
3x2 + 5 = 41
Write original equation.
3x2 = 36
Subtract 5 from each side.
x2 = 12
Divide each side by 3.
x = + 12
Take square roots of each side.
x = + 4 3
Product property
x =
+ 2 3
Simplify.

24. The solutions are – 3 + and – 3 – 35
EXAMPLE 4
Standardized Test Practice
1 5
(z + 3)2 = 7
Write original equation.
(z + 3)2 = 35
Multiply each side by 5.
z + 3 = + 35
Take square roots of each side.
z = – 3 + 35
Subtract 3 from each side.
The solutions are – and – 3 – 35
The correct answer is C.

25. Your Turn: for Examples 3, and 4 5x2 = 80 The solutions are and . 4 –
Solve the equation.
5x2 = 80
ANSWER
The solutions are and . 4 –
z2 – 7 = 29
ANSWER
The solutions are and . 6 –
3(x – 2)2 = 40
The solutions are
2 30 3 2 +
ANSWER

26. EXAMPLE 5
Model a dropped object with a quadratic function
Science Competition
For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground ?

27. EXAMPLE 5 h = – 16t 2 + h0 0 = – 16t 2 + 50 – 50 = – 16t 2 50 16 50 +
Model a dropped object with a quadratic function
SOLUTION
h = – 16t 2 + h0
Write height function.
0 = – 16t
Substitute 0 for h and 50 for h0.
– = – 16t 2
Subtract 50 from each side.
50 16
= t2
Divide each side by – 16. 50 16 +
= t
Take square roots of each side. t
Use a calculator.

28. EXAMPLE 5
Model a dropped object with a quadratic function
ANSWER
Reject the negative solution, – 1.8, because time must be positive. The container will fall for about 1.8 seconds before it hits the ground.

29. GUIDED PRACTICE Your Turn: for Example 5
WHAT IF? In Example 5, suppose the egg container is dropped from a height of 30 feet. How long does the container take to hit the ground?
ANSWER
The container will fall for about 1.4 seconds before it hits the ground.