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10.6
Warm UpWarm Up
Lesson QuizLesson QuizLesson PresentationLesson Presentation
Solve Quadratic Equations by the Quadratic Formula
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10.6 Warm-Up
Evaluate the expression for the given value of x.
1. 15 – (–x) + 9; x = –2
2. 14 – x + 3; x = 8
ANSWER 22
ANSWER 9
ANSWER 22 ft
A basketball is thrown in the air from a height of 6 feet with an initial velocity of 40 feet per second. What is the height (in feet) of the ball after 2 seconds?
3.
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10.6 Example 1
What are the solutions of 3x2 + 5x = 8?
–1 and – A 83 B –1 and 8
3 C 1 and – 83 D 1 and 8
3
SOLUTION
Write original equation.3x2 + 5x = 8
Write in standard form.3x2 + 5x – 8 = 0
Quadratic formula ±x = –b b2 – 4ac2a
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10.6 Example 1
Substitute values in the quadratic formula: a = 3, b = 5, and c = –8.
Simplify.= –5± 1216
Simplify the square root.= –5 ± 116
The solutions of the equation are –5 + 116 = 1 and
–5 – 116 = – 8
3
x = –5 ± 52 – 4(3)(–8)2(3)
The correct answer is C. ANSWER
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10.6 Example 2
2x2 – 7 = x
2x2 – x – 7 = 0
Solve 2x2 – 7 = x.
Write original equation.
Write in standard form.
Quadratic formulax =
b2 – 4ac+ ––b2a
Substitute values in the quadratic formula: a = 2, b = –1, and c = –7.
Simplify.
– (–1) –+ ( –1)2 – 4(2)(–7)
2(2)=
4=+ –1 57
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10.6 Example 2
ANSWER
CHECK
1 + 57 2.14 and 1 – 4
–1.64.4
The solutions are 57
Write the equation in standard form, 2x2 – x – 7 = 0. Then graph the related function y = 2x2 – x – 7. The x-intercepts are about –1.6 and 2.1. So, each solution checks.
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10.6 Guided Practice
Use the quadratic formula to solve the equation. Round your solution to the nearest hundredth, if necessary.
1. x2 – 8x + 16 = 0
ANSWER 4
2. 3n2 – 5n = –1
ANSWER 0.23, 1.43
3. 4z2 = 7z + 2
ANSWER 0.25, 2
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10.6 Example 3
y = 10x2 – 94x + 3900
4200 = 10x2 – 94x – 3900
0 = 10x2 – 94x – 300
Write function.
Substitute 4200 for y.
Write in standard form.
SOLUTION
For the period 1971 2001, the number y of films produced in the world can be modeled by the function y = 10x2 94x + 3900 where x is the number of years since 1971. In what year were 4200 films produced?
FILM PRODUCTION
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10.6 Example 3
Substitute values in the quadratic formula: a = 10, b = –94, and c = –300.
20,83694 + –20
= Simplify.
ANSWER There were 4200 films produced about 12 years after 1971, or in 1983.
x = (–94)2 – 4 (10)(–300)–(–94)
+ –2(10)
The solutions of the equation are:
94 + 20,83620
12 94 – 20,83620
–3.and
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10.6 Guided Practice
WHAT IF? In Example 3, find the year when 4750 films were produced.
4.
ANSWER 1986
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10.6 Example 4
Tell what method you would use to solve the quadraticequation. Explain your choice(s).
a. 10x2 – 7 = 0
SOLUTION
a. The quadratic equation can be solved using square roots because the equation can be written in the form x2 = d.
b. x2 + 4x = 0
b. The equation can be solved by factoring because the expression x2 + 4x can be factored easily. Also, the equation can be solved by completing the square because the equation is of the form ax2 + bx + c = 0 where a = 1 and b is an even number.
c. 5x2 + 9x – 4 = 0
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10.6 Example 4
c. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation can be solved using the quadratic formula.
SOLUTION
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10.6 Guided Practice
Tell what method you would use to solve the quadraticequation. Explain your choice(s).5. x2 + x – 6 = 0
ANSWER Factoring because the expression factors easily.
6. x2 – 9 = 0
ANSWERFactoring because the expression factors easily. Using square roots is another option since the equation can be written in the form x2 = d.
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10.6 Guided Practice
7. x2 + 6x = 5
ANSWERCompleting the square because the equation is of the form ax2 + bx = c where a = 1 and b is an even number. Another method is the quadratic formula since the equation does not factor easily.
Tell what method you would use to solve the quadraticequation. Explain your choice(s).
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10.6 Lesson Quiz
1. 6x2 – 6 = –5x
Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary.
2. 2x2 + 3x – 8 = 0
–2.87, 1.39ANSWER
,ANSWER 32– 2
3
3. 3m2 – m + 9 = 12
ANSWER –0.85, 1.18
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10.6 Lesson Quiz
You have seen that the function y = 10x2 94x + 3900 models the number y of films produced in the world, where x is the number of years since 1971. In what year were 10,000 films produced?
4.
ANSWER about 29.8 years after 1971, or during the year 2000