9-5 Solving Quadratic Equations by Using the Quadratic Formula The solutions are 2.3 and –5.3. Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2

2. x − 10x + 16 = 0

2

6. 5x + 5 = −13x SOLUTION:   Write the equation in standard form.

SOLUTION:   For this equation, a = 1, b = –10, and c = 16. For this equation, a = 5, b = 13, and c = 5.

The solutions are 8 and 2. 2

4. x + 3x = 12 SOLUTION:   Write the equation in standard form.

For this equation, a = 1, b = 3, and c = –12.

The solutions are –0.5 and –2.1. Solve each equation. State which method you used. 2 8. 2x − 3x − 6 = 0 SOLUTION:   Solve using the quadratic formula. For this equation, a = 2, b = –3, and c = –6. Quadratic Formula:

The solutions are 2.3 and –5.3. 2

6. 5x + 5 = −13x SOLUTION:   Write the equation in standard form.

The solutions are 2.6 and –1.1. 2

10. x − 9x = −19 For this equation, a = 5, b = 13, and c = 5.

SOLUTION:   Solve by using the quadratic formula. Write the equation in standard form.

For this equation, a = 1, b = –9, and c = 19. Quadratic Formula: eSolutions Manual - Powered by Cognero

The solutions are –0.5 and –2.1.

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The discriminant is 41. Since the discriminant is positive, the equation has 9-5 Solving Quadratic The solutions are 2.6Equations and –1.1. by Using the Quadratic Formula two real solutions. 2

10. x − 9x = −19 SOLUTION:   Solve by using the quadratic formula. Write the equation in standard form.

2

14. 3x − x = 8 SOLUTION:   Write the equation in standard form.

For this equation, a = 3, b = –1, and c = –8. For this equation, a = 1, b = –9, and c = 19. Quadratic Formula: The discriminant is 97. Since the discriminant is positive, the equation has two real solutions. Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2 16. 4x + 5x − 6 = 0 SOLUTION:   For this equation, a = 4, b = 5, and c = –6. The solutions are 5.6 and 3.4. State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 2 12. 2x − 11x + 10 = 0 SOLUTION:   For this equation, a = 2, b = –11, and c = 10. The solutions are The discriminant is 41. Since the discriminant is positive, the equation has two real solutions. 2

 and –2.

2

18. 6x − 12x + 1 = 0 SOLUTION:   For this equation, a = 6, b = –12, and c = 1.

14. 3x − x = 8 SOLUTION:   Write the equation in standard form.

For this equation, a = 3, b = –1, and c = –8. The solutions are 1.9 and 0.1. The Manual discriminant is 97. eSolutions - Powered by Cognero Since the discriminant is positive, the equation has two real solutions.

2

20. 2x − 5x = −7 SOLUTION:   Write the equation in standard form.

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9-5 Solving Quadratic Equations by Using the Quadratic Formula The solutions are The solutions are 1.9 and 0.1.

.

2

2

20. 2x − 5x = −7

24. 4x = −16x − 16

SOLUTION:   Write the equation in standard form.

SOLUTION:   Write the equation in standard form.

For this equation, a = 2, b = –5, and c = 7.

For this equation, a = 4, b = 16, and c = 16.

The discriminant is negative, so the equation has no real solutions, Ø. 2

22. 81x = 9

The solution is –2.

SOLUTION:   Write the equation in standard form.

2

26. −3x = 8x − 12 SOLUTION:   Write the equation in standard form.

For this equation, a = 81, b = 0, and c = –9.

For this equation, a = –3, b = –8, and c = 12.

The solutions are

.

2

24. 4x = −16x − 16 SOLUTION:   Write the equation in standard form.

For this equation, a = 4, b = 16, and c = 16.

The solutions are –3.7 and 1.1. 28. AMUSEMENT PARKS The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops them vertically 60 feet. A function that 2 approximates this drop is h = −16t + 60, where h is the height in feet and t is the time in seconds. About how many seconds does it take for riders to drop 60 feet? SOLUTION:   2

−16t + 60 = 0 For this equation, a = –16, b = 0, and c = 60.

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It takes about 1.9 seconds for the riders to drop 60 9-5 Solving Quadratic Equations by Using the Quadratic Formula feet. The solutions are –3.7 and 1.1. 28. AMUSEMENT PARKS The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops them vertically 60 feet. A function that 2 approximates this drop is h = −16t + 60, where h is the height in feet and t is the time in seconds. About how many seconds does it take for riders to drop 60 feet?

Solve each equation. State which method you used. 2 30. 3x − 24x = −36 SOLUTION:   Solve by using the quadratic formula. Write the equation in standard form.

SOLUTION:   2

−16t + 60 = 0 For this equation, a = –16, b = 0, and c = 60.

For this equation, a = 3, b = –24, and c = 36. Use the Quadratic Formula.

 

The solutions are 6 and 2. 2

32. 4x + 100 = 0 It takes about 1.9 seconds for the riders to drop 60 feet. Solve each equation. State which method you used. 2 30. 3x − 24x = −36 SOLUTION:   Solve by using the quadratic formula. Write the equation in standard form.

For this equation, a = 3, b = –24, and c = 36. Use the Quadratic Formula.

SOLUTION:   Solve using the quadratic formula. First, check the value of the discriminant. For this equation, a = 4, b = 0, and c = 100.

The discriminant is negative, so the equation has no real solutions, ø. 34. 12 − 12x = −3x

2

SOLUTION:   Solve by factoring. Write the equation in standard form.

Factor.

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For this equation, a = 4, b = 0, and c = 100. The discriminant is 0. The discriminant is negative, so by theUsing equation no Since the discriminant is 0, the equation has one real 9-5 Solving Quadratic Equations thehas Quadratic Formula solution. real solutions, ø. 34. 12 − 12x = −3x

2

SOLUTION:   Solve by factoring. Write the equation in standard form.

40.  SOLUTION:   Write the equation in standard form.

Factor. For this equation, a = 2, b =

, and c =

.

The solution is 2. State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 2

36. 2x − 5x + 20 = 0 SOLUTION:   For this equation, a = 2, b = –5, and c = 20.

The discriminant is –135. Since the discriminant is negative, the equation has no real solutions.

The discriminant is 18.25. Since the discriminant is positive, the equation has two real solutions. 2

42. TRAFFIC The equation d = 0.05v + 1.1v models the distance d in feet it takes a car traveling at a speed of v miles per hour to come to a complete stop. If Hannah’s car stopped after 250 feet on a highway with a speed limit of 65 miles per hour, was she speeding? Explain your reasoning. SOLUTION:   Write the equation in standard form.

2

38. 0.5x − 2x = −2 SOLUTION:   Write the equation in standard form.

For this equation, a = 0.05, b = 1.1, and c = –250.

For this equation, a = 0.5, b = –2, and c = 2.

The discriminant is 0. Since the discriminant is 0, the equation has one real solution. 40.  SOLUTION:   Write the equation in standard form. eSolutions Manual - Powered by Cognero

No, she was not speeding; Sample answer: Hannah was traveling at about 61 mph, so she was not speeding. Without graphing, determine the number ofPage x- 5 intercepts of the graph of the related function for each function.

The discriminant is 0.04. No, she was not speeding; Sample answer: Hannah Since the discriminant is positive, the graph of the 9-5 Solving Quadratic Equations by she Using was traveling at about 61 mph, so wasthe notQuadratic Formula function will have two x-intercepts. speeding. Without graphing, determine the number of xintercepts of the graph of the related function for each function. 44. 

Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2 46. −2x − 7x = −1.5 SOLUTION:   Write the equation in standard form.

SOLUTION:   Write the equation in standard form.

For this equation, a = –2, b = –7, and c = 1.5.

For this equation, a = 1, b =

, and c =

.

The discriminant is 0.04. Since the discriminant is positive, the graph of the function will have two x-intercepts. Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2 46. −2x − 7x = −1.5 SOLUTION:   Write the equation in standard form.

The solutions are –3.7 and 0.2. 2

48. x − 2x = 5 SOLUTION:   Write the equation in standard form.

For this equation, a = 1, b = –2, and c = –5. For this equation, a = –2, b = –7, and c = 1.5.

The solutions are 3.4 and –1.4. The solutions are –3.7 and 0.2. eSolutions Manual - Powered by Cognero

2

48. x − 2x = 5 SOLUTION:  

50. MULTIPLE REPRESENTATIONS In this problem, you will investigate writing a quadratic Page 6 equation with given roots.  2 If p is a root of 0 = ax + bx + c, then (x – p ) is a 2

c. You could write an equation with three roots by multiplying the corresponding factors together and setting it equal to zero. If an equation has the three roots 1, 2, 3, then the corresponding factors would be The solutions are 3.4Equations and –1.4. by Using the Quadratic Formula 9-5 Solving Quadratic (x – 1), (x – 2), and (x – 3). The equation would then be: 50. MULTIPLE REPRESENTATIONS In this problem, you will investigate writing a quadratic equation with given roots.  2 If p is a root of 0 = ax + bx + c, then (x – p ) is a 2

factor of ax + bx + c.

  This is not a quadratic equation since it is of degree 3.      a. Tabular Copy and complete the first two columns of the table. b. Algebraic Multiply the factors to write each equation with integral coefficients. Use the equations to complete the last column of the table. Write each equation. c. Analytical How could you write an equation with three roots? Test your conjecture by writing an equation with roots 1, 2, and 3. Is the equation quadratic? Explain. SOLUTION:   a. For any two roots m and p , in the left hand column, the middle column will be the corresponding factors (x – m), (x – p ).    b. The equation with these factors will be: (x – m)(x 2 – p ) = 0 which simplifies to x – (m + p )x + mp = 0. Use this to fill in the column of the table. 

52. REASONING Use factoring techniques to 2

determine the number of real zeros of f (x) = x − 8x + 16. Compare this method to using the discriminant. SOLUTION:   2

For f(x) = x − 8x + 16, a = 1, b =  −8 and c = 16. 2

2

Then the discriminate is b − 4ac or (−8) −4(1)(16) = 0.  The polynomial can be factored to get f (x) = (x 2. − 4) Solve to find the real zeros.

So the only real zero is 4. The discriminant is 0, so the only real zero is 4. The discriminant is 0, so there is 1 real zero. The discriminant tells us how many real zeros there are. Factoring tells us what they are. CCSS STRUCTURE  Determine whether there are two, one, or no real solutions. 54. The graph of a quadratic function touches but does not cross the x-axis. SOLUTION:   If the graph is tangent to the x-axis, meaning there is only one x-intercept, then there is only one real solution.

c. You could write an equation with three roots by multiplying the corresponding factors together and setting it equal to zero. If an equation has the three roots 1, 2, 3, then the corresponding factors would be (x – 1), (x – 2), and (x – 3). The equation would then be:

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56. Both a and b are greater than 0 and c is less than 0 in a quadratic equation. SOLUTION:   2

The discrimininant is b – 4ac. No matter the value 2 of b, b  will always be positive. If  a is greater than 0 and c is less than 0, then – 4ac will be positive. Thus Page 7 the discrimininant would be positive. So there would  be two real solutions.

leading coefficient has to be 1 and the x - and xSOLUTION:   term must be isolated. It is also easier if the If the graph is tangent to the x-axis, meaning there is coefficient of the x-term is even; if not, the only one x-intercept, then there is only one real calculations become harder when dealing with 9-5 Solving 2 solution.Quadratic Equations by Using the Quadratic Formula fractions.  For example x + 4x = 7 can be solved by completing the square.  56. Both a and b are greater than 0 and c is less than 0 in a quadratic equation. SOLUTION:   2

The discrimininant is b – 4ac. No matter the value 2 of b, b  will always be positive. If  a is greater than 0 and c is less than 0, then – 4ac will be positive. Thus the discrimininant would be positive. So there would  be two real solutions. 58. WRITING IN MATH Describe the advantages and disadvantages of each method of solving quadratic equations. Which method do you prefer, and why?

  Quadratic Formula: The Quadratic Formula will work for any quadratic equation and exact solutions can be found. This method can be time consuming, especially if an equation is easily factored.  For example, use the  2 Quadratic Formula to find the solutions of f (x) = 4x + 13 x + 5. 

SOLUTION:   Factoring: Factoring is easy if the polynomial is factorable and complicated if it is not. Not all equations are factorable.  2

2

For example f (x) = x – 8x + 16 factors to (x – 4) . 2 However, f (x) = x – 16x + 8 can not be factored.

 

 

Graphing: Graphing only gives approximate answers, but it is easy to see the number of solutions. Using square roots is easy when there is no x-term. 

See students’ preferences. 60. SHORT RESPONSE The triangle shown is an isosceles triangle. What is the value of x?

2

For example, for the quadratic f (x) = 2x – 17x + 4, you can see the two solutions in the graph. However, it will be difficult to identify the solution x = 8.2578049 in the graph.  SOLUTION:   Because an isosceles triangle has two equal angles, x could be equal to 64, or it could be equal to the unnamed angle, where 180 = 2x + 64.

[-5, 15] scl: 2 by [-30, 10] scl: 4

 

The value of x is 58 or 64.

Completing the square: Completing the square can be used for any quadratic equation and exact solutions can be found, but the 2 leading coefficient has to be 1 and the x - and xterm must be isolated. It is also easier if the coefficient of the x-term is even; if not, the calculations become harder when dealing with 2

fractions.  For example x + 4x = 7 can be solved by completing the square. 

62. What are the solutions of the quadratic equation 6h + 6h = 72? A 3 or −4 B −3 or 4 C no solution D 12 or −48

2

SOLUTION:   Write the equation in standard form.

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For this equation, a = 6, b = 6, and c = –72.

9-5 Solving Quadratic Equations by Using the Quadratic Formula The solutions are 1.6 or 7.4. The value of x is 58 or 64. 62. What are the solutions of the quadratic equation 6h + 6h = 72? A 3 or −4 B −3 or 4 C no solution D 12 or −48

2

Describe the transformations needed to obtain the graph of g(x) from the graph of f (x). 2

66. f (x) = 4x 2 g(x) = 2x SOLUTION:   2

The graph of g(x) = ax stretches or compresses the

SOLUTION:   Write the equation in standard form.

2

graph of f (x) = 4x vertically. The change in a is