171S3.2p Quadratic Equations, Functions, Zeros, & Models

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CHAPTER 3: Quadratic Functions and Equations; Inequalities

3.1 The Complex Numbers3.2Quadratic 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 two quadratic functions of same shape using sliders and shows two forms of equation. http://cfcc.edu/mathlab/geogebra/quadratic2ss.html

This program graphs one quadratic function using sliders and shows the roots (solutions) and two forms of the equation. http://cfcc.edu/mathlab/geogebra/quadratic_roots.html

Solve Quadratic Equation by using this Excel program. http://cfcc.edu/faculty/cmoore/quadratic_formula.xls

This program graphs one or two quadratic functions and shows two forms of equation, roots, axis of symmetry, and vertex for each graph.http://cfcc.edu/mathlab/geogebra/quadratic_graph_rootsa.html

Oct 2­2:47 PM

Oct 2­8:47 AM Oct 2­8:53 AM

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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

Sep 28­7:25 AM

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.

Sep 27­3:17 PM

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.

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:

Sep 27­3:17 PM

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.

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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

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?

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

Sep 27­3:17 PM

Example (continued)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.

Sep 27­9:10 PM

255/2. Solve. (5x – 2)( 2x + 3) = 0

255/6. Solve. 10x2 – 16x + 6 = 0

Sep 30­2:26 PM

256/12. Solve. 4x2 + 12 = 0

256/18. Solve. 3t3 + 2t = 5t2

Sep 27­9:11 PM

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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256/20. Solve. 3x3 + x2 ­ 12x ­ 4 = 0(Hint: Factor by grouping.)

256/19. Solve. 7x3 + x2 ­ 7x ­ 1 = 0(Hint: Factor by grouping.)

Sep 27­9:12 PM

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.

Sep 27­9:13 PM

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

Sep 27­9:18 PM

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

Sep 27­9:22 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

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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

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

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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

257/104. Solve (3x + 2)2 + 7(3x + 2) ­ 8 = 0

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257/106. Solve 12 = (m2 ­ 5m)2 + (m2 ­ 5m)

257/106. Solve 12 = (m2 ­ 5m)2 + (m2 ­ 5m)

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?

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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? http://www.stlouisarch.com/experience/the­gateway­arch/journey­to­the­top/