Standard 8
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Transcript of Standard 8
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Standard 8 Solve a quadratic equation
Solve 6(x – 4)2 = 42. Round the solutions to the nearesthundredth.
6(x – 4)2 = 42 Write original equation.
(x – 4)2 = 7 Divide each side by 6.
x – 4 = ± 7 Take square roots of each side.
7 x = 4 ± Add 4 to each side.
ANSWER
The solutions are 4 + 6.65 and 4 – 1.35.7 7
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Solve a quadratic equation
Standard 8
CHECKTo check the solutions, first write the equation so that 0 is on one side as follows: 6(x – 4)2 – 42 = 0. Then graph the related function y = 6(x – 4)2 – 42. The x-intercepts appear to be about 6.6 and about 1.3. So, each solution checks.
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EXAMPLE 1 Solve quadratic equations
Solve the equation. Round the solution to the nearest hundredth if necessary.
1. 2(x – 2)2 = 18
GUIDED PRACTICE Solve a Quadratic Equation
ANSWER –1, 5
2. 4(q – 3)2 = 28 ANSWER 0.35, 5.65
3. 3(t + 5)2 = 24 ANSWER –7.83, –2.17
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EXAMPLE 1 Solve quadratic equations
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.
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Solve quadratic equationsEXAMPLE 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.
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Solve quadratic equationsEXAMPLE 1
c. b2 + 12 = 5Write original equation.
b2 = – 7 Subtract 12 from each side.
ANSWER
Negative real numbers do not have real square roots. So, there is no solution.
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EXAMPLE 2 Take square roots of a fraction
Solve 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.
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Take square roots of a fractionEXAMPLE 2
ANSWER
The solutions are – and 32
32
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Approximate solutions of a quadratic equation
EXAMPLE 3
Solve 3x2 – 11 = 7. Round the solutions to the nearesthundredth.
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.
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Approximate solutions of a quadratic equation
EXAMPLE 3
x ± 2.45 Use a calculator. Round to the nearesthundredth.
ANSWER
The solutions are about – 2.45 and about 2.45.
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EXAMPLE 1 Solve quadratic equations
Solve the equation.
1. c2 – 25 = 0
GUIDED PRACTICE
ANSWER –5, 5.
2. 5w2 + 12 = – 8 ANSWER no solution
3. 2x2 + 11 = 11 ANSWER 0
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EXAMPLE 1 Solve quadratic equations
Solve the equation.
4. 25x2 = 16
GUIDED PRACTICE
ANSWER 4 5
4 5– ,
5. 9m2 = 100 ANSWER 103– ,
103
6. 49b2 + 64 = 0 ANSWER no solution
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EXAMPLE 1 Solve quadratic equations
Solve the equation. Round the solutions to the nearest hundredth.
7. x2 + 4 = 14
GUIDED PRACTICE
ANSWER – 3.16, 3.16
8. 3k2 – 1 = 0 ANSWER – 0.58, 0.58
9. 2p2 – 7 = 2 ANSWER – 2.12, 2.12
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Standard 8 Complete the square
Find the value of c that makes the expression x2 + 5x + c a perfect square trinomial. Then write the expression as the square of a binomial.
STEP 1Find the value of c. For the expression to be a perfect square trinomial, c needs to be the square of half the coefficient of bx.
Find the square of half the coefficient of bx.
22 = 25
4c = 5
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Standard 8 Complete the square
STEP 2Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial.
Substitute 25 for c.4
x2 + 5x + c = x2 + 5x +254
Square of a binomial52
2+x =
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GUIDED PRACTICE
Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
1. x2 + 8x + c ANSWER 16; (x + 4)2
2. x2 12x + c
3. x2 + 3x + c
ANSWER 36; (x 6)2
ANSWER ; (x )294
32
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EXAMPLE 2 Solve a quadratic equation
Solve x2 – 16x = –15 by completing the square.
SOLUTION
Write original equation.x2 – 16x = –15
Add , or (– 8)2, to each side.
– 16 2
2x2 – 16x + (– 8)2 = –15 + (– 8)2
Write left side as the square of a binomial.
(x – 8)2 = –15 + (– 8)2
Simplify the right side.(x – 8)2 = 49
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EXAMPLE 2 Standardized Test Practice
Take square roots of each side.x – 8 = ±7
Add 8 to each side.x = 8 ± 7
ANSWER
The solutions of the equation are 8 + 7 = 15 and 8 – 7 = 1.
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EXAMPLE 2 Standardized Test Practice
CHECK
You can check the solutions in the original equation.
If x = 15:
(15)2 – 16(15) –15 ?=
–15 = –15
If x = 1:
(1)2 – 16(1) –15 ?=
–15 = –15
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EXAMPLE 3 Solve a quadratic equation in standard form
Solve 2x2 + 20x – 8 = 0 by completing the square.
SOLUTION
Write original equation.2x2 + 20x – 8 = 0
Add 8 to each side.2x2 + 20x = 8
Divide each side by 2.x2 + 10x = 4
Add 10 2
2, or 52, to each side.x2 + 10x + 52 = 4 + 52
Write left side as the square of a binomial.
(x + 5)2 = 29
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EXAMPLE 3 Solve a quadratic equation in standard form
Take square roots of each side.x + 5 =
± 29
Subtract 5 from each side.x = –5 ± 29
ANSWER
The solutions are – 5 + 29 0.39 and – 5 - 29 –10.39.
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GUIDED PRACTICE
4. x2 – 2x = 3
ANSWER 1, 3
5. m2 + 10m = –8
ANSWER 9.12, 0.88
6. 3g2 – 24g + 27 = 0
ANSWER 1.35, 6.65