96 The Quadratic Formula and the Discriminant discriminant (noun) dih skrim uh nunt Definition: The discriminant of the equation a x2 + b x + c = 0 is the value of the related expression b 2 − 4 a c . Main Idea: The value of the discriminant tells how many realnumber solutions a quadratic equation has. Word Origin: The word discriminant comes from the Latin “discriminare,” which means “to distinguish.” recall that quadratic equations can have two, one or no real number solutions. a quadratic equation can never have more than two solutions. Essential Understanding You can find the solution(s) of any quadratic equation using the quadratic formula . Be sure to write a quadratic equation in standard form before using the quadratic formula. Use the quadratic formula to solve each equation. A.) 2x2 + 5x + 3 = 0 B.) 4x2 + 7x − 15 = 0 C.) 18x2 − 45x − 50 = 0 D.) 3x2 + 19x = 154 E.) 5x2 − 47x = 156 When the radicand in the quadratic formula is not a perfect square, you can use a calculator to approximate the solutions of an equation. Use the quadratic formula to solve each equation. round your answer to the nearest hundredth. F.) 5x2 + 12x − 2 = 0 G.) 8x2 − 7x − 5 = 0 2 H.) 3x + 5x = 4 There are many methods for solving quadratic equation. method When to Use Graphing Use if you have a graphing calculator handy. Square roots Use if the equations has no xterm. Factoring Use if you can factor the equation easily. completing the square Use if the coefficient of x2 is 1, but you cannot easily factor the equation. Quadratic formula Use if the equation cannot be factored easily or at all. There are many methods for solving a quadratic equation. You can always use the quadratic formula, but sometimes another method may be easier. Which method(s) would you choose to solve each equation? Justify your reasoning. I.) x2 + 4x − 15 = 0 2 J.) 4x − 41x = 73 K.) x2 + 4x − 60 = 0 Quadratic equations can have two, one, or no realnumber solutions. Before you solve a quadratic equation. you can determine how many real number solutions is has by using the discriminant. The discriminant is the expression under the radical sign in the quadratic formula. x= the discriminant −b±√b2 −4ac ⇐ 2a The discriminant of a quadratic equation can be positive, zero, or negative. Find the number of realnumber solutions of each equation. L.) x2 − 2x + 3 = 0 M.) x2 + 3x + 11 = 0 N.) x2 + 2x = 0 Use and method to solve each equation. If necessary, round your answer to the nearest hundredth. O.) 3w2 = 48 P.) 6g2 − 18 = 0 Q.) k2 − 4k =− 4
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