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171S3.2q Quadratic Equations, Functions, Zeros, and Models
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February 21, 2013
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CHAPTER 3: Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers3.2 Quadratic Equations, Functions, Zeros, and Models3.3 Analyzing Graphs of Quadratic Functions3.4 Solving Rational Equations and Radical Equations3.5 Solving Equations and Inequalities with Absolute Value
MAT 171 Precalculus AlgebraDr. Claude Moore
Cape Fear Community College
This program graphs quadratic functions and shows two forms of equation. http://cfcc.edu/mathlab/geogebra/quadratic2ss.html
This program graphs a quadratic function and shows the roots (solutions). http://cfcc.edu/mathlab/geogebra/quadratic_roots.html
Solve Quadratic Equation by using this Excel program. http://cfcc.edu/faculty/cmoore/quadratic_formula.xls
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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
Finding Zeros of Quadratic Function by Graphing with TI Calculator
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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 formf (x) = ax2 + bx + c, a ≠ 0, where a, b, and c are real numbers.
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 imaginarynumber 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.
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Example
SolutionSolve 2x2 x = 3.
Example Checking the Solutions
Check: x = – 1
TRUE
TRUE
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EquationSolving Principles
The Principle of Square Roots: If x2 = k, then
Completing the Square
• 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 firstdegree 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.
To solve a quadratic equation by completing the square:
171S3.2q Quadratic Equations, Functions, Zeros, and Models
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Example
Solve 2x2 − 1 = 3x.
Solution
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Quadratic Formula
The solutions of ax2 + bx + c = 0, a ≠ 0, are given by
This formula can be used to solve any quadratic equation.
DiscriminantWhen 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.
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Applications
Some applied problems can be translated to quadratic equations.
ExampleTime 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?
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.
This program graphs a quadratic function and shows the roots (solutions). http://cfcc.edu/mathlab/geogebra/quadratic_roots.html
Solve Quadratic Equation by using this Excel program. http://cfcc.edu/faculty/cmoore/quadratic_formula.xls
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255/2. Solve. (5x – 2)( 2x + 3) = 0
255/6. Solve. 10x2 – 16x + 6 = 0
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256/12. Solve. 4x2 + 12 = 0
256/18. Solve. 3t3 + 2t = 5t2
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256/20. Solve. 3x3 + x2 12x 4 = 0(Hint: Factor by grouping.)
256/19. Solve. 7x3 + x2 7x 1 = 0(Hint: Factor by grouping.)
171S3.2q Quadratic Equations, Functions, Zeros, and Models
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256/20. Solve. 3x3 + x2 12x 4 = 0(Hint: Factor by grouping.)
256/19. Solve. 7x3 + x2 7x 1 = 0(Hint: Factor by grouping.)
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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.
256/24. 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.
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256/26. 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.
256/28. 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.
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256/30. Solve by completing the square to obtain exact solutions.x2 + 8x = 15
256/32. Solve by completing the square to obtain exact solutions.x2 = 22 + 10x
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256/34. Solve by completing the square to obtain exact solutions.x2 + 6x + 13 = 0
256/36. Solve by completing the square to obtain exact solutions.2x2 5x 3 = 0
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256/44. Use the quadratic formula to find exact solutions.x2 + 1 = x
256/48. Use the quadratic formula to find exact solutions.2t2 5t = 1
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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
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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
256/58. Find discriminant and describe solutions.4x2 12x + 9 = 0
256/62. Find discriminant and describe solutions.5t2 4t = 11
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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
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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
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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
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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
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257/92. Solve x4 + 3 = 4x2
257/96. Solve y4 15y2 16 = 0
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257/100. Solve t2/3 + t1/3 6 = 0
257/102. Solve x1/2 4x1/4 = 3
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257/104. Solve (3x + 2)2 + 7(3x + 2) 8 = 0
257/106. Solve 12 = (m2 5m)2 + (m2 5m)
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257/106. Solve 12 = (m2 5m)2 + (m2 5m)
257/104. Solve (3x + 2)2 + 7(3x + 2) 8 = 0
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Taipei 101 TowerTaipei, Taiwan 1670 fthttp://video.about.com/architecture/WorldsTallestBuildingsTheTaipei101Tower.htm
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?
171S3.2q Quadratic Equations, Functions, Zeros, and Models
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http://www.gatewayarch.com/experience/thegatewayarch/
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 from the top to reach the ground?
Sep 302:06 PM
Taipei 101 TowerTaipei, Taiwan 1670 fthttp://video.about.com/architecture/WorldsTallestBuildingsTheTaipei101Tower.htm
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?
Sep 302:06 PM
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 from the top to reach the ground?
http://www.gatewayarch.com/experience/thegatewayarch/
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.
