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Quadratic equations Lesson 2 Source : Lial, Hungerford and Holcomb (2007), Mathematics with Applications, 9th edition, Pearson Prentice Hall, ISBN (Chapter 1 pp.56-64)

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Quadratic Equations Quadratic equations could be of two types: 1.Complete Where a, b, and c are real numbers and a 0 2. Incomplete:

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Quadratic Equations: Incomplete Quadratic Equations Solving Incomplete Quadratic Equations:

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Quadratic Equations: Incomplete Quadratic Equations Solving Incomplete Quadratic Equations:

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Quadratic Equations: Complete Quadratic Equations Solving Complete Quadratic Equations: Three methods to solve: 1. Using Discriminant 2. Viete’s Theorem 3. Completing the square

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Quadratic Equations: Complete Quadratic Equations Solving Complete Quadratic Equations: 1.Using Discriminant:

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3. Discriminant can be negative and we'd get no real solutions. The "discriminat" tells us what type of solutions we'll have. 1. Discriminant can be positive and we'd get two unequal real solutions 2. Discriminant can be zero and we'd get one solution (called a repeated or double root because it would gives us two equal real solutions). Quadratic Equations: Complete Quadratic Equations

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Example 3. Discriminant can be negative and we'd get no real solutions. Feel the power of the formula! Example 1. Discriminant can be positive and we'd get two unequal real solutions Example 2. Discriminant can be zero and we'd get one solution (called a repeated or double root because it would gives us two equal real solutions). Quadratic Equations: Complete Quadratic Equations

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Quadratic Equations: Complete Quadratic Equations We will complete the square First get the constant term on the other side We are now going to add a number to the left side so it will factor into a perfect square. 36 Solving Complete Quadratic Equations: 2. Completing Square

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This can be written as Now we'll get rid of the square by square rooting both sides. Remember you need both the positive and negative root! Add 6 to both sides to get x alone.

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Quadratic Equations: Complete Quadratic Equations Example: Solve by completing the square:

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Quadratic Equations: Complete Quadratic Equations Solution:

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Quadratic Equations: Complete Quadratic Equations According to Viete’s theorem if x 1 and x 2 are the solutions of then Solving Complete Quadratic Equations: 3.Viete’s theorem

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Quadratic Equations: Complete Quadratic Equations Example 1: Solve by Viete’s theorem: Example 2: Solve by Viete’s theorem:

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Quadratic Equations: Complete Quadratic Equations Example 1: Solve by Viete’s theorem:

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Quadratic Equations: Complete Quadratic Equations Example 2: Solve by Viete’s theorem:

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Solving by reducing to quadratic equations: Example 1: Reduce to quadratic equation and solve Practice yourself: Reduce to quadratic equation and solve Some higher degree equations could be reduced to quadratic equations and then solved

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Solving by reducing to quadratic equations: Example 1: Solution

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Important! If we know the roots of quadratic equation we could factorize any quadratic expression easy way: Example 2: Solve by Viete’s theorem:

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Factorizing using roots Example 1: Factorizing using roots Solution: Factorizing:

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Factorizing using roots Example 2: Factorizing using roots Solution: Factorizing:

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