Concept Category 4. Quadratic Equations
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1 Concept Category 4 Quadratic Equations
2 1 Solving Quadratic Equations by the Square Root Property
3 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
4 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
5 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
6 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
7 Square Root Property Example Solve (3x 17) = 8 3x 17 = x 17 7 x Martin-Gay, Developmental Mathematics 7
8 Solving Quadratic Equations by the Quadratic Formula
9 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
10 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
11 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) Martin-Gay, Developmental Mathematics 11
12 Two kinds of answers: Decimal Answers (to graph): and 0.9 and 0.1 Simplified Radical Answers (SAT, ACT, and other college placement exams): Martin-Gay, Developmental Mathematics 1
13 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
14 Martin-Gay, Developmental Mathematics 14
15 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
16 Quadratic Formula SAT example Example 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) or, 10 or Martin-Gay, Developmental Mathematics 16
17 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
18 Radicals = = 4 = 5 = 10 = 1 Martin-Gay, Developmental Mathematics 18
19 Perfect Squares Martin-Gay, Developmental Mathematics 19
20 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 = 4* = 0 = 4*5 = 5 3 = 16* = 4 75 = 5*3 = = 4*10 = 10 Martin-Gay, Developmental Mathematics 0
21 Simplify each expression A] B] Martin-Gay, Developmental Mathematics 1
22 + To combine radicals: combine the coefficients of like radicals Martin-Gay, Developmental Mathematics
23 Simplify each radical first, then combine * 3 16 * *5 3* Martin-Gay, Developmental Mathematics 3
24 Practice NOW * * 3 3* 3 3 5* Martin-Gay, Developmental Mathematics 4
25 * Multiply the coefficients and then multiply the radicands and then simplify Martin-Gay, Developmental Mathematics 5
26 Multiply and then simplify 5 * * * *14 6* * *10 80 Martin-Gay, Developmental Mathematics 6
27 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
28 X 3 = X * X = X X Y 5 = Y 4 Y = Y Y Martin-Gay, Developmental Mathematics 8
29 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
30 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
31 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
32 5 5 * *3 7 *3 7 *3 7 81* w 8 w 4 * 8w w 4 8w 3x 3 * 3 x x 4 9x 1x Martin-Gay, Developmental Mathematics 3
33 Practice NOW x 16x 4 Martin-Gay, Developmental Mathematics 33
34 SOLUTIONS x Martin-Gay, Developmental Mathematics 34
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