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1.29.13 7.3 The Discriminant Tuesday, July 17, 2012 Vocabulary • Discriminant: the part of the quadratic formula under the radical, often denoted by D D = b2 - 4ac It allows us to determine the nature of the roots of a quadratic Remember, the roots of a quadratic are calculated as follows: Calculate the discriminant, then calculate the roots: HA2T-A Page 1 HA2T-A Page 2 The Nature of the Roots of a Quadratic Equation: Let ax2 + bx + c = 0 be a quadratic equation with real coefficients 1. If b2 - 4ac > 0, there are 2 unequal roots 2. If b2 - 4ac = 0, there is one real double root 3. If b2 - 4ac < 0, there are 2 conjugate imaginary roots Test for Rational Roots If a quadratic has integral coefficients and its discriminant is a perfect square, then the equation has rational roots. If a quadratic equation can be transformed into an equivalent equation that meets this test, then it has rational roots Methods for Solving Quadratic Equations: 1. Factoring if the discriminant is a perfect square 2. Completing the Square if the equation has the form: x2 + (even number)x + constant = 0 3. Quadratic Formula always works, but may be more time consuming 4. Graphing if the discriminant is > 0. Will not work with imaginary roots Word Problems: The volume of a rectangular jewelry box 4cm in height is 136 cm3. If the perimeter of the base is 24 cm, find the dimensions of the base. HA2T-A Page 3 A 5 in. by 7 in. photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame. A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides. The cut-off squares are 5 cm on a side. If the volume of the box is 100 cm3, find the dimensions of the original piece of metal. Equations in Quadratic Form Tuesday, July 17, 2012 10:28 PM 7-4 Equations in Quadratic Form Vocabulary • Quadratic Form: any equation that can be written: Where a is not equal to 0 and f(x) is some function of x. To solve: replace f(x) with a single variable. Examples: HA2T-A Page 4 Practice: Homework: MathXL 7.3-7.4 HW HA2T-A Page 5