Concept Category 4 Quadratic Equations
1 Solving Quadratic Equations by the Square Root Property
Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce additional methods for solving quadratic equations. Square Root Property If b is a real number and a = b, then a b Martin-Gay, Developmental Mathematics 3
Square Root Property Example Solve x = 49 x 49 7 Solve x = 4 x = x Solve (y 3) = 4 y 3 4 y = 3 y = 1 or 5 Martin-Gay, Developmental Mathematics 4
Square Root Property Example Solve x + 4 = 0 x = 4 There is no real solution because the square root of 4 is not a real number. Martin-Gay, Developmental Mathematics 5
Square Root Property Example Solve (x + ) = 5 x 5 5 x = ± 5 x = + 5 or x = 5 x = 3 or x = 7 Martin-Gay, Developmental Mathematics 6
Square Root Property Example Solve (3x 17) = 8 3x 17 = 8 7 3 x 17 7 x 17 3 7 Martin-Gay, Developmental Mathematics 7
Solving Quadratic Equations by the Quadratic Formula
The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation. Martin-Gay, Developmental Mathematics 9
The Quadratic Formula A quadratic equation written in standard form, ax + bx + c = 0, has the solutions. x b b 4ac a Martin-Gay, Developmental Mathematics 10
The Quadratic Formula Example Solve 11n 9n = 1 by the quadratic formula. 11n 9n 1 = 0 set one side = 0 a = 11, b = -9, c = -1 n 9 ( 9) 4(11)( 1) (11) 9 81 44 9 15 Martin-Gay, Developmental Mathematics 11
Two kinds of answers: Decimal Answers (to graph): 9 15 9 11. 9 11. 9 11. and 0.9 and 0.1 Simplified Radical Answers (SAT, ACT, and other college placement exams): 9 5 5 9 5 5 Martin-Gay, Developmental Mathematics 1
Practice: Solve x using QFormula Present your answers in Decimals and Simplified Radicals: a] f ( x) 4x 1x 63 b] y x 1x 46 Martin-Gay, Developmental Mathematics 13
Martin-Gay, Developmental Mathematics 14
1/18 Practice Now: a f x x x b g x x ] ( ) 10 1 ] ( ) ( 3) 5 Vertex point? x-intercept points? y-intercept point? *Vertex point? *x-intercept points? *y-intercept point? Martin-Gay, Developmental Mathematics 15
Quadratic Formula SAT example Example 1 8 5 Solve x + x = 0 by the quadratic formula. x + 8x 0 = 0 multiply both sides by 8 a = 1, b = 8, c = 0 x 8 (8) 4(1)( 0) (1) 8 64 80 8 144 8 1 0 4 or, 10 or Martin-Gay, Developmental Mathematics 16
Concept Category 4 Quadratics Standard Form of Quadratic Equation b Vertex : x int : Factoring or Quadratic Formula a Vertex Form of Quadratic Equation Vertex: transformation first Radical Operations Solving Radical Equations Nth Roots Complex Numbers Martin-Gay, Developmental Mathematics 17
Radicals 4 16 5 100 144 = = 4 = 5 = 10 = 1 Martin-Gay, Developmental Mathematics 18
Perfect Squares 1 4 9 16 64 81 100 11 5 56 89 34 5 36 49 144 400 169 196 65 Martin-Gay, Developmental Mathematics 19
LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 = 4* = 0 = 4*5 = 5 3 = 16* = 4 75 = 5*3 = 5 3 40 = 4*10 = 10 Martin-Gay, Developmental Mathematics 0
Simplify each expression A] B] 6 7 5 7 3 7 8 7 5 6 3 7 4 7 6 3 6 7 7 Martin-Gay, Developmental Mathematics 1
+ To combine radicals: combine the coefficients of like radicals Martin-Gay, Developmental Mathematics
Simplify each radical first, then combine. 50 3 3 5* 3 16 * *5 3* 4 10 1 Martin-Gay, Developmental Mathematics 3
Practice NOW 3 7 5 48 3 9 * 3 5 16 * 3 3* 3 3 5* 4 3 9 3 0 3 9 3 Martin-Gay, Developmental Mathematics 4
* Multiply the coefficients and then multiply the radicands and then simplify Martin-Gay, Developmental Mathematics 5
Multiply and then simplify 5 * 35 175 *7 5 5 7 8 *3 7 56 6 6 4*14 6* 14 1 14 5 *4 0 8 100 8*10 80 Martin-Gay, Developmental Mathematics 6
X = X Y 6 = Y 3 P 4 X 6 Y = P X 3 Y 4X 4 Y = X Y 8 5C D 10 = 5C 4 D 5 Martin-Gay, Developmental Mathematics 7
X 3 = X * X = X X Y 5 = Y 4 Y = Y Y Martin-Gay, Developmental Mathematics 8
PX 3 Y 3 = X Y * PXY = XY PXY 1X 7 Y = Y 5 8 5C D 9 = 5 Y Martin-Gay, Developmental Mathematics 9
Happy Wed. 1/5 Did everyone have a chance to work on yesterday s SMC practice test? Placement Exams for English and Math are required for ALL CA colleges ( or 4 yr) This practice test have other parts: nd part of Algebra, Geometry, and Precalculus The less you score and more courses ($$$$ + Time) you will need to make up for Martin-Gay, Developmental Mathematics 30
CA College Placement Exams v.s. CAASP Why they all have similar questions? Because your HIGH SCHOOL standards/state Exams are mostly set by public colleges Martin-Gay, Developmental Mathematics 31
5 5 * 5 5 5 4 3 7 3 7 *3 7 *3 7 *3 7 81*49 3969 4 8w 8 w 4 * 8w 4 8 64w 4 8w 3x 3 * 3 x x 4 9x 1x Martin-Gay, Developmental Mathematics 3
Practice NOW 3 5 4 0 3 45 4 3 4 3 x 16x 4 Martin-Gay, Developmental Mathematics 33
SOLUTIONS 3 5 316x Martin-Gay, Developmental Mathematics 34