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**Chapter 7 Quadratic Equations**

TMAT 103 Chapter 7 Quadratic Equations

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**§7.1 Solving Quadratic Equations by Factoring**

TMAT 103 §7.1 Solving Quadratic Equations by Factoring

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**§7.1 – Solving Quadratic Equations by Factoring**

Quadratic Equation – general form: Key principle – Zero Factor Property: If ab = 0, then either a = 0, b = 0, or both

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**§7.1 – Solving Quadratic Equations by Factoring**

Solving a Quadratic Equation by Factoring (b 0) If necessary, write the equation in the form ax2 + bx + c = 0 Factor the nonzero side of the equation Using the preceding problem, set each factor that contains a variable equal to zero Solve each resulting linear equation Check

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**§7.1 – Solving Quadratic Equations by Factoring**

Examples – Solve the following by factoring x2 – 6x + 8 = 0 2x2 + 9x = 5 x – 2x2 = 0

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**§7.1 – Solving Quadratic Equations by Factoring**

Solving a Quadratic Equation by Factoring (b = 0) If necessary, write the equation in the form ax2 = c Divide each side by a Take the square root of each side Simplify the result, if possible

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**§7.1 – Solving Quadratic Equations by Factoring**

Examples – Solve the following by factoring 4x2 = 9 16 – x2 = 0

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**§7.2 Solving Quadratic Equations by Completing the Square**

TMAT 103 §7.2 Solving Quadratic Equations by Completing the Square

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**§7.2 – Solving Quad Equations by Completing the Square**

Solving a Quadratic Equation by Completing the Square The coefficient of the second-degree term must equal (positive) 1. If not, divide each side of the equation by its coefficient Write an equivalent equation in the form x2 + px = q. Add the square of ½ of the coefficient of the linear term to each side; that is, (½p)2 The left side is now a perfect square trinomial. Rewrite the left side as a square Take the square root of each side Solve for x and simplify, if possible Check

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**§7.2 – Solving Quad Equations by Completing the Square**

Examples – Solve the following by completing the square x2 – 6x + 8 = 0 2x2 + 9x = 5 x – 2x2 = 0

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**§7.3 The Quadratic Formula**

TMAT 103 §7.3 The Quadratic Formula

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**§7.3 The Quadratic Formula**

The general quadratic equation can now be solved by completing the square This will generate a formula that can be used to solve any quadratic equation x will be written in terms of a, b, and c

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**§7.3 The Quadratic Formula**

Solving a Quadratic Equation using the Quadratic Formula If necessary, write the equation in the form ax2 + bx + c = 0 Substitute a, b, and c into the quadratic formula Solve for x Check

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**§7.3 The Quadratic Formula**

Examples – Solve the following by using the quadratic formula x2 – 6x + 8 = 0 2x2 + 9x = 5 x – 2x2 = 0

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**§7.3 The Quadratic Formula**

Consider the quadratic formula The discriminant provides insight into the nature of the solutions discriminant

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**§7.3 The Quadratic Formula**

Discriminant If b2 – 4ac > 0, there are 2 real solutions If b2 – 4ac is also a perfect square they are both rational If b2 – 4ac is not a perfect square, they are both irrational If b2 – 4ac = 0, there is only one rational solution If b2 – 4ac < 0, there are two imaginary solutions Chapter 14

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**§7.3 The Quadratic Formula**

Examples – How many and what types of solutions do each of the following have? x2 – 2x + 17 = 0 x2 – x – 2 = 0 x2 + 6x + 9 = 0 2x2 + 2x + 14 = 0

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TMAT 103 §7.4 Applications

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**§7.4 Applications Examples**

The work done in Joules in a circuit varies with time in milliseconds according to the formula w = 8t2 – 12t Find t in ms when w = 16J. A rectangular sheet of metal 24 inches wide is formed into a rectangular trough with an open top and no ends. If the cross-sectional area is 70 in2, find the depth of the trough.

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**§14.1 Complex Numbers in Rectangular Form**

TMAT 103 §14.1 Complex Numbers in Rectangular Form

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**§14.1 – Complex Numbers in Rectangular Form**

Imaginary Unit In mathematics, i is used In technical math, i denotes current, so j is used to denote an imaginary number Rectangular Form of a Complex Number a is the real component, and bj is the imaginary component

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**§14.1 – Complex Numbers in Rectangular Form**

Examples – Express in terms of j and simplify

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**§14.1 – Complex Numbers in Rectangular Form**

Powers of j j = j j2 = –1 j3 = –j j4 = 1 j5 = j j6 = –1 j7 = –j j8 = 1 … Process continues Powers of j evenly divisible by four are equal to 1

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**§14.1 – Complex Numbers in Rectangular Form**

Examples – Express in terms of j and simplify

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**§14.1 – Complex Numbers in Rectangular Form**

Additional Information Complex numbers are not ordered “Greater than” and “Less than” do not make sense Conjugate The conjugate of (a + bj) is (a – bj)

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**§14.1 – Complex Numbers in Rectangular Form**

Addition and subtraction Complex numbers can be added and subtracted as if they were 2 ordinary binomials (a + bj) + (c + dj) = (a + c) + (b + d)j (a + bj) – (c + dj) = (a – c) + (b – d)j

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**§14.1 – Complex Numbers in Rectangular Form**

Examples – Perform the indicated operation (1 – 2j) + (3 – 5j) (–3 + 13j) – (4 – 7j) (½ – 11j) – (½ – 4j)

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**§14.1 – Complex Numbers in Rectangular Form**

Multiplication Complex numbers can be multiplied as if they were 2 ordinary binomials (a + bj)(c + dj) = (ac – bd) + (ad + bc)j

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**§14.1 – Complex Numbers in Rectangular Form**

Examples – Multiply (1 – 2j)(3 – 5j) (–3 + 13j)(4 – 7j) (½ – 11j)(½ – 4j)

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**§14.1 – Complex Numbers in Rectangular Form**

Division Complex numbers can be divided by multiplying numerator and denominator by the conjugate of the denominator

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**§14.1 – Complex Numbers in Rectangular Form**

Examples – Divide

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**§14.1 – Complex Numbers in Rectangular Form**

Solving quadratic equations with a negative discriminant 2 complex solutions Always occur in conjugate pairs Use quadratic formula, or other techniques

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**§14.1 – Complex Numbers in Rectangular Form**

Examples – Solve using the quadratic formula x2 + x + 1 = 0 x2 + 9 = 0

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**§14.1 – Complex Numbers in Rectangular Form**

Adding complex numbers graphically

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**§14.1 – Complex Numbers in Rectangular Form**

Subtracting complex numbers graphically

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