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171S3.2q Quadratic Equations, Functions, Zeros, and Models
February 21, 2013
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 3: Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and Models
3.3 Analyzing Graphs of Quadratic Functions
3.4 Solving Rational Equations and Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value
Solve Quadratic Equation by using this Excel program. http://cfcc.edu/faculty/cmoore/quadratic_formula.xls
This program graphs a quadratic function and shows the roots (solutions). http://cfcc.edu/mathlab/geogebra/quadratic_roots.html
This program graphs quadratic functions and shows two forms of equation. http://cfcc.edu/mathlab/geogebra/quadratic2ss.html
Sep 27­3:17 PM
3.2 Quadratic Equations, Functions, Zeros, and Models
• Find zeros of quadratic functions and solve quadratic equations by using the principle of zero products, by using the principle of square roots, by completing the square, and by using the quadratic formula.
• Solve equations that are reducible to quadratic.
• Solve applied problems using quadratic equations.
Graphing Quadratic Function with TI Calculator
Feb 19­10:15 AM
Quadratic Equations
A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, a ≠ 0,
where a, b, and c are real numbers.
A quadratic equation written in this form is said to be in standard form. A quadratic function f is a function that can be written in the form
f (x) = ax2 + bx + c, a ≠ 0, where a, b, and c are real numbers.
Finding Zeros of Quadratic Function by Graphing with TI Calculator
The zeros of a quadratic function f (x) = ax2 + bx + c are the solutions of the associated quadratic equation ax2 + bx + c = 0. Quadratic functions can have real­number or imaginary­
number zeros and quadratic equations can have real­number or imaginary­number solutions.
Equation­Solving Principles
The Principle of Zero Products:
If ab = 0 is true, then a = 0 or b = 0,
and if a = 0 or b = 0,
then ab = 0.
Sep 27­3:17 PM
Sep 27­3:17 PM
Example
Equation­Solving Principles
Solve 2x2 ­ x = 3.
Solution
The Principle of Square Roots:
If x2 = k, then
Completing the Square
To solve a quadratic equation by completing the square:
Example ­ Checking the Solutions
Check: x = – 1
TRUE
• Isolate the terms with variables on one side of the equation and arrange them in descending order.
• Divide by the coefficient of the squared term if that coefficient is not 1.
• Complete the square by taking half the coefficient of the first­
degree term and adding its square on both sides of the equation.
• Express one side of the equation as the square of a binomial.
• Use the principle of square roots.
• Solve for the variable.
TRUE
Sep 27­3:17 PM
Sep 27­3:17 PM
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171S3.2q Quadratic Equations, Functions, Zeros, and Models
February 21, 2013
Quadratic Formula
The solutions of ax2 + bx + c = 0, a ≠ 0, are given by
Example
This formula can be used to solve any quadratic equation.
Solve 2x2 − 1 = 3x.
Discriminant
Solution
When you apply the quadratic formula to any quadratic equation, you find the value of b2 ­ 4ac, which can be positive, negative, or zero. This expression is called the discriminant.
For ax2 + bx + c = 0, where a, b, and c are real numbers:
b2 ­ 4ac = 0 One real­number solution;
b2 ­ 4ac > 0 Two different real­number solutions;
b2 ­ 4ac < 0 Two different imaginary­number solutions, complex conjugates.
Equations Reducible to Quadratic
Some equations can be treated as quadratic, provided that we make a suitable substitution. Example: x4 ­ 5x2 + 4 = 0 Knowing that x4 = (x2)2, we can substitute u for x2 and the resulting equation is then u2 ­ 5u + 4 = 0. This equation can then be solved for u by factoring or using the quadratic formula. Then the substitution can be reversed by replacing u with x2, and solving for x. Equations like this are said to be reducible to quadratic, or quadratic in form.
Sep 27­3:17 PM
Applications
Sep 27­3:17 PM
255/2. Solve. (5x – 2)( 2x + 3) = 0
Some applied problems can be translated to quadratic equations. Example
Time of Free Fall. The Petronas Towers in Kuala Lumpur, Malaysia are 1482 ft tall. How long would it take an object dropped from the top to reach the ground?
255/6. Solve. 10x2 – 16x + 6 = 0
1. Familiarize. The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds.
2. Translate. Substitute 1482 for s in the formula: 1482 = 16t2.
3. Carry out. Use the principle of square roots.
4. Check. In 9.624 seconds, a dropped object would travel a distance of 16(9.624)2, or about 1482 ft. The answer checks.
5. State. It would take about 9.624 sec for an object dropped from the top of the Petronas Towers to reach the ground.
Solve Quadratic Equation by using this Excel program. http://cfcc.edu/faculty/cmoore/quadratic_formula.xls
This program graphs a quadratic function and shows the roots (solutions). http://cfcc.edu/mathlab/geogebra/quadratic_roots.html
Sep 27­3:17 PM
256/12. Solve. 4x2 + 12 = 0
256/18. Solve. 3t3 + 2t = 5t2 Sep 30­2:26 PM
Sep 27­9:10 PM
256/19. Solve. 7x3 + x2 ­ 7x ­ 1 = 0
(Hint: Factor by grouping.)
256/20. Solve. 3x3 + x2 ­ 12x ­ 4 = 0
(Hint: Factor by grouping.)
Sep 27­9:11 PM
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171S3.2q Quadratic Equations, Functions, Zeros, and Models
256/19. Solve. 7x3 + x2 ­ 7x ­ 1 = 0
(Hint: Factor by grouping.)
February 21, 2013
256/22. Use the graph to find
(a) The x­intercepts
(b) The zeros of the function
The x­intercepts should be written as ordered pairs (x, 0).
The zeros (or roots, solutions) are x­values when y = 0.
256/20. Solve. 3x + x ­ 12x ­ 4 = 0
(Hint: Factor by grouping.)
3
2
256/24. Use the graph to find
(a) The x­intercepts
(b) The zeros of the function
The x­intercepts should be written as ordered pairs (x, 0).
The zeros (or roots, solutions) are x­values when y = 0.
Sep 27­9:11 PM
256/26. Use the graph to find
(a) The x­intercepts
(b) The zeros of the function
Sep 27­9:12 PM
256/30. Solve by completing the square to obtain exact solutions.
x2 + 8x = ­15
The x­intercepts should be written as ordered pairs (x, 0).
The zeros (or roots, solutions) are x­values when y = 0.
256/28. Use the graph to find
(a) The x­intercepts
(b) The zeros of the function
256/32. Solve by completing the square to obtain exact solutions.
x2 = 22 + 10x
The x­intercepts should be written as ordered pairs (x, 0).
The zeros (or roots, solutions) are x­values when y = 0.
Sep 27­9:13 PM
256/34. Solve by completing the square to obtain exact solutions.
x2 + 6x + 13 = 0
Sep 27­9:15 PM
256/44. Use the quadratic formula to find exact solutions.
x2 + 1 = x
256/36. Solve by completing the square to obtain exact solutions.
2x2 ­ 5x ­ 3 = 0
256/48. Use the quadratic formula to find exact solutions.
2t2 ­ 5t = 1
Sep 27­9:15 PM
Sep 27­9:18 PM
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171S3.2q Quadratic Equations, Functions, Zeros, and Models
256/52. Use the quadratic formula to find exact solutions.
5x2 + 3x = 1
256/56. Use the quadratic formula to find exact solutions.
3x2 + 3x = ­4
Sep 27­9:18 PM
256/58. Find discriminant and describe solutions.
4x2 ­ 12x + 9 = 0
February 21, 2013
256/58. Find discriminant and describe solutions.
4x2 ­ 12x + 9 = 0
256/62. Find discriminant and describe solutions.
5t2 ­ 4t = 11
Sep 27­9:21 PM
256/74. Find the zeros of the function algebraically and solve graphically. Round solutions to 3­decimal places if not integer.
f(x) = 3x2 + 8x + 2
256/62. Find discriminant and describe solutions.
5t2 ­ 4t = 11
Sep 27­9:21 PM
256/78. Find the zeros of the function algebraically and solve graphically. Round solutions to 3­decimal places if not integer.
f(x) = x2 ­ x ­ 4
Sep 27­9:22 PM
Sep 27­9:22 PM
256/82. Find the zeros of the function algebraically and solve graphically. Round solutions to 3­decimal places if not integer.
f(x) = 3x2 + 5x + 1
Sep 27­9:22 PM
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171S3.2q Quadratic Equations, Functions, Zeros, and Models
257/84. Find the zeros of the function algebraically and solve graphically. Round solutions to 3­decimal places if not integer.
f(x) = 4x2 ­ 4x ­ 5
February 21, 2013
257/92. Solve x4 + 3 = 4x2
257/96. Solve y4 ­ 15y2 ­ 16 = 0
Sep 27­9:27 PM
257/100. Solve t2/3 + t1/3 ­ 6 = 0
Sep 27­9:27 PM
257/104. Solve (3x + 2)2 + 7(3x + 2) ­ 8 = 0
257/106. Solve 12 = (m2 ­ 5m)2 + (m2 ­ 5m)
257/102. Solve x ­ 4x = ­3
1/2
1/4
Sep 27­9:27 PM
257/104. Solve (3x + 2)2 + 7(3x + 2) ­ 8 = 0
257/106. Solve 12 = (m2 ­ 5m)2 + (m2 ­ 5m)
Sep 27­9:27 PM
Time of a Free Fall. The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds. Use this formula for Exercises 111 and 112.
258/111. The Taipei 101 Tower, also known as the Taipei Financial Center, in Taipei, Taiwan, is 1670 ft tall. How long would it take an object dropped from the top to reach the ground? Taipei 101 Tower
Taipei, Taiwan 1670 ft
http://video.about.com/archite
cture/World­s­Tallest­
Buildings­­The­Taipei­101­
Tower.htm
Sep 27­9:27 PM
Sep 30­2:06 PM
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171S3.2q Quadratic Equations, Functions, Zeros, and Models
Time of a Free Fall. The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds. Use this formula for Exercises 111 and 112.
258/112. At 630 ft, the Gateway Arch in St. Louis is the tallest man­made monument in the United States. How long would it take an object dropped http://www.gatewayarch.com/experien
from the top to reach the ce/the­gateway­arch/
February 21, 2013
Time of a Free Fall. The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds. Use this formula for Exercises 111 and 112.
258/111. The Taipei 101 Tower, also known as the Taipei Financial Center, in Taipei, Taiwan, is 1670 ft tall. How long would it take an object dropped from the top to reach the ground? ground? Taipei 101 Tower
Taipei, Taiwan 1670 ft
http://video.about.com/archite
cture/World­s­Tallest­
Buildings­­The­Taipei­101­
Tower.htm
Sep 30­2:06 PM
Sep 30­2:06 PM
Time of a Free Fall. The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds. Use this formula for Exercises 111 and 112.
258/112. At 630 ft, the Gateway Arch in St. Louis is the tallest man­made monument in the United States. How long would it take an object dropped http://www.gatewayarch.com/experienc
e/the­gateway­arch/
from the top to reach the ground? Sep 30­2:06 PM
6