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9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
1. 3x +(– 6x)
Warm-UpSimplify the expression.
2. 5 + 4x + 2
–3 xANSWER
4x + 7ANSWER
3. 4(2x – 1) + x 9x – 4ANSWER
4. – (x + 4) – 6 x – 7x – 4ANSWER
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root. Warm-Up
Simplify the expression.
13. (x5)3
14. (– x)3
11. (3xy)3 27x3y3ANSWER
12. xy2 xy3 x2y5ANSWER
x15ANSWER
–x3ANSWER
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Find the two square roots of each number.
7 is a square root, since 7 • 7 = 49.
–7 is also a square root, since –7 • –7 = 49.
10 is a square root, since 10 • 10 = 100.
–10 is also a square root, since –10 • –10 = 100.
49 = –7–
49 = 7
100 = 10
100 = –10–
A. 49
B. 100
C. 225
15 is a square root, since 15 • 15 = 225.225 = 15
225 = –15– –15 is also a square root, since –15 • –15 = 225.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
A. 25
Check It Out: Example 1
5 is a square root, since 5 • 5 = 25.–5 is also a square root, since –5 • –5 = 25.
12 is a square root, since 12 • 12 = 144.
–12 is also a square root, since –12 • –12 = 144.
25 = –5–25 = 5
144 = 12
144 = –12–
Find the two square roots of each number.
B. 144
C. 289289 = 17
289 = –17–
17 is a square root, since 17 • 17 = 289.
–17 is also a square root, since –17 • –17 = 289.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Additional Example 3A: Evaluating Expressions Involving Square RootsSimplify the expression.
Evaluate the square root.
Add.= 25
Multiply.= 18 + 7
3 36 + 7
3 36 + 7 = 3(6) + 7
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Additional Example 3B: Evaluating Expressions Involving Square RootsSimplify the expression.
+ 25 16
3 4
25 16
3 4
+3 4 = +1.5625
Evaluate the square roots.
= 1.25 + 3 4
25 16
= 1.5625.
= 2 Add.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Check It Out: Example 3ASimplify the expression.
Evaluate the square root.
Add.= 14
Multiply.= 10 + 4
2 25 + 4
2 25 + 4 = 2(5) + 4
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Check It Out: Example 3BSimplify the expression.
+ 18 t2
1 4
18 t2
1 4
+ 1 4
= + 9
Evaluate the square roots.= 3 + 1 4
18 t2
= 9.
= 3 Add.1 4
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
3. Evaluate the expression.
A. 17
B. 17
C. 19
D. 72
Lesson Quiz for Student Response Systems
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
4. Evaluate the expression.
A. 4
B. 8
C. 16
D. 40
Lesson Quiz for Student Response Systems
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
A Quadratic equation is an equation that can be written in the following standard form:
ax2 + bx + c = 0 where a does not equal 0
If b = 0ax2 + c = 0
These are the type we will work with today.
If b = 0 and c = 0ax2 = 0
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solving Quadratic Equations
a.x2 = 4
b.x2 = 5
c.x2 = 0
d.x2 = -1
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solving Quadratic Equations
a.x2 = 25
b.x2 = 7
c.x2 = 81
d.x2 = -12
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve the equation.a. 2x2 = 8
SOLUTION
a. 2x2 = 8Write original equation.
x2 = 4 Divide each side by 2.
x = ± 4 = ± 2 Take square roots of each side. Simplify.
ANSWER The solutions are – 2 and 2.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
EXAMPLE 1
b. m2 – 18 = – 18 Write original equation.
m2 = 0 Add 18 to each side..
The square root of 0 is 0.m = 0
ANSWER
The solution is 0.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
EXAMPLE 1c. b2 + 12 = 5
Write original equation.
b2 = – 7 Subtract 12 from each side.
ANSWER
Negative real numbers do not have real square roots. So, there is no solution.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve quadratic equationsSolve the equation.1. c2 – 25 = 0
SOLUTION
c2 – 25 = 0 Write original equation.
c = ± 25 = ± 5 Take square roots of each side. Simplify.
GUIDED PRACTICE
ANSWER
The solutions are – 5 and 5.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
EXAMPLE 2
Take square roots of a fractionSolve 4z2 = 9.
SOLUTION
4z2 = 9. Write original equation.
z2 = 94 Divide each side by 4.
Take square roots of each side.z = ± 94
z = ± 32
Simplify.
ANSWER
The solutions are – and 32
32
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Approximate solutions of a quadratic equation Solve 3x2 – 11 = 7. Round the solutions to the nearest
hundredth.
SOLUTION
3x2 – 11 = 7 Write original equation.
3x2 = 18 Add 11 to each side.
x2 = 6 Divide each side by 3.
x = ± 6 Take square roots of each side.
x ± 2.45 Use a calculator. Round to the nearesthundredth.
The solutions are about – 2.45 and about 2.45.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
EXAMPLE 1
Solve quadratic equationsSolve the equation.2. 5w2 + 12 = – 8
SOLUTION
5w2 + 12 = – 8 Write original equation.
w = –4 Take square roots of each side. Simplify.
GUIDED PRACTICE
ANSWER
Negative real numbers do not have a real square root.So there is no solution.
5w2 = – 8 –12 Subtract 12 from each side.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve quadratic equationsSolve the equation.3. 2x2 + 11 = 11
SOLUTION
2x2 + 11 = 11 Write original equation.
x = 0 The root of 0 is 0.
GUIDED PRACTICE
ANSWER
The solution is 0 .
2x2 = 0 Subtract 11 from each side.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.EXAMPLE 1
Solve quadratic equationsSolve the equation.4. 25x2 = 16
SOLUTION
25x2 = 16 Write original equation.
Take square roots of each side.
GUIDED PRACTICE
Divided each to by 25.x =1625
x = ± 1625
x = ± 4 5
Simplify.
ANSWER
The solution is – and . 4 5
4 5
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve quadratic equationsSolve the equation.5. 9m2 = 100
SOLUTION
9m2 = 100 Write original equation.
Take square roots of each side.
GUIDED PRACTICE
Divided each to by 9.m =100 9
m = ± 100 9
m = ± 10 3
Simplify.
ANSWER
The solution is – and . 10 3
10 3
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.EXAMPLE 1
Solve quadratic equationsSolve the equation.6. 49b2 + 64 = 0
SOLUTION
49b2 + 64 = 0 Write original equation.
Take square roots of each side.
GUIDED PRACTICE
Divided each to by 9.
49b2 = – 64
b2 = –64 49
Subtract 64 from each side.
ANSWERNegative real numbers do have real square root. So there is no solution.
b = – 6449
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve quadratic equationsSolve the equation. Round the solution to the nearest hundredth.
7. x2 + 4 = 14
SOLUTION
x2 + 4 = 14 Write original equation.
Take square roots of each side.
GUIDED PRACTICE
Use a calculation. Round to the nearest hundredth.
x2 = 10 Subtract 4 from each side.
ANSWER The solutions are about – 3.16 and 3.16.
x = 10+–
x = +– 3.16
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve quadratic equationsSolve the equation. Round the solution to the nearest hundredth.
8. 3k2 – 1 = 0
SOLUTION
3k2 – 1 = 0 Write original equation.
Take square roots of each side.
GUIDED PRACTICE
Use a calculation. Round to the nearest hundredth.
3k2 = 1 Add 1 to each side.
k = +– 0.58
k2 = 1 3
k = +– 13
Divided each to by 3.
The solutions are about – 0.58 and 0.58.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solve the equation. Round the solution to the nearest hundredth.
9. 2p2 – 7 = 2
SOLUTION
2p2 – 7 = 2 Write original equation.
Take square roots of each side.
Use a calculation. Round to the nearest hundredth.
2p2 = 2 + 7 Add 7 to each side.
p = +– 2.12
p2 = 9 2
p = +– 92
Divided each to by 2.
GUIDED PRACTICE
The solutions are about – 2.12 and 2.12.
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solving Quadratic Equations
a.x2 + 5 = 21
b.x2 – 2 = 7
c.2x2 = 18
d.3x2 = 75
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solving Quadratic Equations
a.2x2 -8 = 0
b.x2 +25 = 0
c.x2 - 1.44 = 0
d.5x2 = -15
9.1 – Students will be able to evaluate square roots. Students will be able to solve a quadratic equation by finding the square root.
Solving Quadratic Equations
a.3x2 -48 = 0
b.120 - 6x2 = -30
c.12x2 - 60 = 0