Download Solve the equation. a - PMS-Math

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.
1.
3x +(– 6x)
ANSWER
–3 x
2.
5 + 4x + 2
ANSWER
4x + 7
3.
4(2x – 1) + x
ANSWER
9x – 4
4.
– (x + 4) – 6 x
ANSWER
– 7x – 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.
Warm-Up
Simplify the expression.
11. (3xy)3
ANSWER
27x3y3
12.
xy2 xy3
ANSWER
x2 y5
13.
(x5)3
ANSWER
x15
ANSWER
–x3
14. (–
x)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.
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.
A. 49
–
49 = 7
49 = –7
B. 100
100 = 10
– 100 = –10
C. 225
–
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.
225 = 15
15 is a square root, since 15 • 15 = 225.
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.
Check It Out: Example 1
Find the two square roots of each number.
A. 25
–
25 = 5
25 = –5
B. 144
144 = 12
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.
– 144 = –12 –12 is also a square root, since
–12 • –12 = 144.
C. 289
289 = 17
–
17 is a square root, since 17 • 17 = 289.
289 = –17 –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 Roots
Simplify the expression.
3 36 + 7
3 36 + 7 = 3(6) + 7
Evaluate the square root.
= 18 + 7
Multiply.
= 25
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.
Additional Example 3B: Evaluating Expressions
Involving Square Roots
Simplify the expression.
25 + 3
16
4
25 + 3 =
16
4
3
1.5625 +
4
25 = 1.5625.
16
= 1.25 + 3
4
Evaluate the
square roots.
=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 3A
Simplify the expression.
2
25 + 4
2 25 + 4 = 2(5) + 4
Evaluate the square root.
= 10 + 4
Multiply.
= 14
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 3B
Simplify the expression.
18 + 1
t2
4
18 + 1 = 9 + 1
t2
4
4
18 = 9.
t2
=3+ 1
4
Evaluate the square roots.
= 31
4
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.
Lesson Quiz for Student Response Systems
3. Evaluate the expression.
A. 17
B. 17
C. 19
D. 72
9.1 – Students will be able to evaluate square roots. Students will be able
to solve a quadratic equation by finding the square root.
Lesson Quiz for Student Response Systems
4. Evaluate the expression.
A. 4
B. 8
C. 16
D. 40
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 = 0
ax2 + c = 0
These are the type we will work
with today.
If b = 0 and c = 0
ax2 = 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 = 8
x2 = 4
x = ± 4 = ± 2
ANSWER
Write original equation.
Divide each side by 2.
Take square roots of each side.
Simplify.
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
m2 = 0
m= 0
ANSWER
The solution is 0.
Write original equation.
Add 18 to each side..
The square root of 0 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 1
2
c. b + 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.
GUIDED PRACTICE
Solve quadratic equations
Solve the equation.
1. c2 – 25 = 0
SOLUTION
c2 – 25 = 0
c = ± 25 = ± 5
ANSWER
The solutions are – 5 and 5.
Write original equation.
Take square roots of each side.
Simplify.
9.1 – Students will be able to evaluate square roots. Students will be able
to solve a quadratic equation by finding the square root.
Take square roots of a fraction
EXAMPLE 2
Solve 4z2 = 9.
SOLUTION
4z2 = 9.
Write original equation.
9
z2 = 4
z = ±
Divide each side by 4.
9
4
z=± 3
2
ANSWER
The solutions are – 3 and 3
2
2
Take square roots of each side.
Simplify.
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
3x2 = 18
x2 = 6
x = ± 6
x
± 2.45
Write original equation.
Add 11 to each side.
Divide each side by 3.
Take square roots of each side.
Use a calculator. Round to the nearest
hundredth.
The solutions are about – 2.45 and about 2.45.
9.1 – Students will be able to evaluate square roots. Students will be able
1 equation by finding the square root.
to EXAMPLE
solve a quadratic
Solve the equation.
2. 5w2 + 12 = – 8
Solve quadratic equations
GUIDED PRACTICE
SOLUTION
5w2 + 12 = – 8
5w2 = – 8 –12
w =  –4
Write original equation.
Subtract 12 from each side.
Take square roots of each side.
Simplify.
ANSWER
Negative real numbers do not have a real square root.
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 the equation.
3. 2x2 + 11 = 11
Solve quadratic equations
GUIDED PRACTICE
SOLUTION
2x2 + 11 = 11
2x2 = 0
x=0
ANSWER
The solution is 0 .
Write original equation.
Subtract 11 from each side.
The root of 0 is 0.
9.1 – Students will be able to evaluate square roots. Students will be able
1 equation by finding the square root.
to EXAMPLE
solve a quadratic
Solve quadratic equations
GUIDED PRACTICE
Solve the equation.
4. 25x2 = 16
SOLUTION
25x2 = 16
16
x = 25
x = ±  16
25
x=± 4
5
Write original equation.
Divided each to by 25.
Take square roots of each side.
Simplify.
ANSWER
The solution is – 4
5
and 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 equations
Solve the equation.
GUIDED PRACTICE
2
5. 9m = 100
SOLUTION
9m2 = 100
100
m= 9
m = ± 100
9
m = ± 10
3
Write original equation.
Divided each to by 9.
Take square roots of each side.
Simplify.
ANSWER
The solution is – 10 and 10 .
3
3
9.1 – Students will be able to evaluate square roots. Students will be able
1 equation by finding the square root.
to EXAMPLE
solve a quadratic
Solve quadratic equations
Solve the equation.
GUIDED PRACTICE
2
6. 49b + 64 = 0
SOLUTION
49b2 + 64 = 0
49b2 = – 64
–64
2
b = 49
b =– 64
49
ANSWER
Write original equation.
Subtract 64 from each side.
Divided each to by 9.
Take square roots of each side.
Negative real numbers do have real square
root. 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.
GUIDED PRACTICE Solve quadratic equations
Solve the equation. Round the solution to the nearest
hundredth.
7.
x2 + 4 = 14
SOLUTION
x2 + 4 = 14
x2 = 10
Subtract 4 from each side.
x = +
– 10
Take square roots of each side.
x = +
– 3.16
ANSWER
Write original equation.
Use a calculation. Round to
the nearest hundredth.
The solutions are about – 3.16 and 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 equations
GUIDED PRACTICE
Solve the equation. Round the solution to the nearest
hundredth.
8.
3k2 – 1 = 0
SOLUTION
3k2 – 1 = 0
3k2 = 1
1
= 3
k = +
– 1
3
k = +
– 0.58
k2
Write original equation.
Add 1 to each side.
Divided each to by 3.
Take square roots of each side.
Use a calculation. Round to
the nearest hundredth.
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.
GUIDED PRACTICE
Solve the equation. Round the solution to the nearest
hundredth.
9.
2p2 – 7 = 2
SOLUTION
2p2 – 7 = 2
2p2 = 2 + 7
9
2
p = 2
p = +
– 9
2
p = +
– 2.12
Write original equation.
Add 7 to each side.
Divided each to by 2.
Take square roots of each side.
Use a calculation. Round to
the nearest hundredth.
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