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Section 10.5 – Page 506 Objectives Use the quadratic formula to find solutions to quadratic equations. Use the quadratic formula to find the zeros of a quadratic function. Evaluate the discriminant to determine how many real roots a quadratic equation has and whether it can be factored.

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Glossary Terms 10.5 The Quadratic Formula discriminant quadratic formula

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Rules and Properties The Quadratic Formula 10.5 The Quadratic Formula x = –b b 2 – 4ac 2a2a For ax 2 + bx + c = 0, where a 0: The Discriminant b 2 – 4ac

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Use the quadratic Formula to solve x² - 10 + 3x = 0 x² + 3x – 10 = 0 Rewrite in standard form a = _, b = _, c = _ Identify the coefficients -(3) ± √(3)² - 4(1)(-10) 2(1) Substitute coefficients into quadratic formula -3 ± √ 49 2 -3 ± 7 2 -3 + 7 2 = 2= 2 -3 – 7 2 = -5 1 3 -10 So the solutions are 2 and –5.

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Find the zeroes of y = 3x² + 2x - 4 a = ___, b = ___, c = ___ 32 -4 -2 ± √2² - 4(3)(-4) 2(3) -2 ± √4 – (-48) 6 -2 ± √52 6 -2 + √52 6 ≈ 0.87 -2 - √52 6 ≈ -1.54 So the solutions are 0.87 and –1.54

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The Discriminant The discriminant of a quadratic equation allows you to determine how many real-number solutions the quadratic equation has. b² - 4ac If the value of the discriminant is less than 0, the equation has no real solutions (b² - 4ac < 0) If the value of the discriminant is equal to 0, the equation has exactly one real solution (b² - 4ac = 0) If the value of the discriminant is greater than 0, the equation has two real solutions

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Find the number of real solutions to each of the following quadratic equations 3x² - 2x + 1 = 0 a = __, b = __, c = __ 3 -21 (-2)² - 4(3)(1) ? 0 4 – 12 ? 0 -8 __ 0 < So there are no real solutions 4x = 4x² + 1 4x² - 4x + 1 = 0 a = __, b = __, c = __ 4 -4 1 (-4)² - 4(4)(1) ? 0 16 – 16 ? 0 0 __ 0 = So there is exactly one real solution

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If the discriminant is a perfect square, the equation can be factored. Determine whether 6x² + 23x + 30 can be factored a = 6, b = 23, c = 120 23² - 4(6)(20) 529 - 480 49 49 is a perfect square so the equation can be factored Use the quadratic formula to find the factors x = -23 ± √ (23)² - 4(6)(20) 2(6) x = -23 ± √ 49 12 x = -23 + 7 12 = -16/12 = -4/3 X = -23 – 7 12 = 30/12 = -5/2

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Factored Form – Cont. x = - 4/3 x = -5/2 3x = -4 3x + 4 = 0 2x = -5 2x + 5 = 0 So the factored form is (3x + 4)(2x + 5) = 0 You can check by doing FOIL to see if you get the original equation

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Assignment Page 510 # 16 – 36, 45 – 48, 52, 53

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