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3.3 QUADRATIC EQUATIONS & INEQUALITIES QUIZ Tell true or false of the following statement: given linear functions f(x) and g(x), if (fg)(x) = 0, then either f(x) = 0 or g(x) = 0. QUADRATIC EQUATIONS Solve quadratic equations by factoring Solve by square root method Solve by the quadratic formula. QUADRATIC EQUATION IN ONE VARIABLE Quadratic equation in standard form: SOLVE BY FACTORING Fact: Given two functions f(x) and g(x), if (fg)(x) = 0, then either f(x) = 0 or g(x) = 0. 3y(y - 1) = 0 X2 – 49 = 0 X2 + 11x = 12 6p2 – 5p = 6 2x2 + 4x -16 = 0 X2 -6x + 9 =0 SQUARE ROOT METHOD Fact: Given a function f(x) , if ( f(x) )2 = B, B ≥ 0, then f(x) = ± √ B. x2 = 49 (4x - 3)2 = 36 (1 - z)2 = 8 (a + 2)2 = -5 QUADRATIC FORMULA The solutions of the quadratic equation ax2 + bx + c = 0, where a ≠ 0, are given by the formula x= - b ± √ b2 – 4ac x2 + 5x + 7 = 0 3p2 + 7p -2 = 0 (x - 3)(x + 5) = -7 2a DISCRIMINANT b2 – 4ac is called the discriminant of the quadratic equation Value of b2 – 4ac Number of Solutions Type of Solutions Positive Two Real 0 One(identical solution) Real Negative Two Nonreal complex DISCRIMINANT AND GRAPH Value of b2 – 4ac Number of Real Solutions No. of x-intercepts Positive Two Two 0 One(identical solution) One Negative None None SOLVING QUADRATIC INEQUALITIES a>0 x1 x2 If x1 < x < x2, then f(x) < 0 If x = x1 or x2, then f(x) = 0 If x < x1 or x > x2, then f(x)> 0 a<0 x x1 x2 If x1 < x < x2, then f(x) > 0 If x = x1 or x2, then f(x) = 0 If x < x1 or x > x2, then f(x)< 0 x SOLVING QUADRATIC INEQUALITIES 1. Solve the corresponding quadratic equation Finding the x- intercepts of the function 2. Identify the intervals determined by the solutions of the equation 3. Use a test value from each interval to determine which intervals from the solution set SOLVING QUADRATIC INEQUALITIES x2 + 4x -12 ≥ 0 x(x-7) < 8 x2 + 4x +4 < 0 x2 + 6x +9 > 0 2x2 + 4x +4 < 0 -x2 + 4x - 6 > 0 HOMEWORK PG. 189: 5 – 50(M5),51,55,60 PG. 190: 63,66,68, 72 – 93(M3), 95 – 103(odds) KEY: 30, 45, 51, 95 Reading: 3.4 Modeling Reminder: Project 2 Skill test 2 Exam 2 Due March 2nd Due March 4th March 7th,8th