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
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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
A] Divide the coefficients, divide the radicands if possible, B] rationalize the denominator so that the denominator is always an integer! Martin-Gay, Developmental Mathematics 35
Fractional Radicands 3 4 3 9 Simplify exponents/radicals first. Then reduce. 4 9 4 3 9 0 36 0 36 4 5 5 5 6 6 3 Martin-Gay, Developmental Mathematics 36
56 7 8 4* Martin-Gay, Developmental Mathematics 37
You Try: 45 16 45 9 5 3 5 16 16 4 15 49 15 15 49 7 63 63 9 7 3 7 6 7 7 81 81 9 9 9 3 Martin-Gay, Developmental Mathematics 38
How about. 50 3 5 3 BIG NO NO! When simplifying radicals, there can never be a radical in the denominator of the final answer. Rationalization: the steps to change a radical denominator to a whole number. Martin-Gay, Developmental Mathematics 39
In the world of SAT, ACT, math placement exams: Method: 4 4 * 6 6 8 4 * 7 7 Multiply the Denominator by the same radical 6 * 7 7 7 4 4 49 7 Fraction expansion Martin-Gay, Developmental Mathematics 40
This can be divided which leaves the radical in the denominator. 5 1 * 10 We do not leave radicals in the denominator. So we need to rationalize. Martin-Gay, Developmental Mathematics 41
3 3 1 * 3 3 3 Rationalize 3 3 9 3 3 6 Reduce the fraction. 3 Martin-Gay, Developmental Mathematics 4
Try it yourself. Simplify the following. 1) 98 7 7 7 7 7 7 7 7 14 49 7 14 14 7 How about with variables 6 x ) x3 3 y y y x 3 y y y y x y 3 y y 3 x yy y x 3 y y Martin-Gay, Developmental Mathematics 43
1 3 4 * 3 3 4 4x x 5 4 x x x x x x x x 3x 18x 4x 13 3 3 x *3x x 6 x x 9 4 x 5 3 3xy 8x y 1x 6 4 3xy 4 *7x xy y 4 *3x Martin-Gay, Developmental Mathematics 44 6???
Happy MONDAY! Quick Check on Wedn. This week: Will be on gradebook *Standard Form: vertex point, x and y intercepts, sketch *Vertex Form: vertex point, x and y intercepts, sketch *Radicals: Simplify, add, subtr, multi, div (rationalization) *Solving Radical equations Martin-Gay, Developmental Mathematics 45
Quick Check Tomorrow Martin-Gay, Developmental Mathematics 46
Self-Evaluation, Correction, Practice 1) Self Evaluation: Mark right or wrong with. CO Conceptual Errors i.e. wrong formulas, wrong solving steps CA computational Errors ) Write the actual correction work on separate paper Turn in (1) and () today 3) Practice: use the other version Martin-Gay, Developmental Mathematics 47
Simplify 5 6 5 3 5 6 5 3 5 3 5 3 5 3 5 6 5 18 59 When radicals have binomial radicals in the denominator, multiply the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator. Simplify by distributing or using FOIL. then continue to completely simplify. 3 9 4 5 Be sure to simplify all numbers outside the radical if they all have common factors. Martin-Gay, Developmental Mathematics 48
Rationalizing Binomial Radicals 6 3 3 6 3 7 3 5 4 3 1 3 3 4 5 15 13 Martin-Gay, Developmental Mathematics 49
The Quadratic Formula Example Solve x(x + 6) = 30 by the quadratic formula. x + 6x + 30 = 0 a = 1, b = 6, c = 30 x 6 (6) (1) 4(1)(30) 6 36 10 6 84 This is Complex Number!!!! Martin-Gay, Developmental Mathematics 50
The Discriminant The expression under the radical sign in the formula (b 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. Martin-Gay, Developmental Mathematics 51
The Discriminant Example Use the discriminant to determine the number and type of solutions for the following equation. 5 4x + 1x = 0 a = 1, b = 4, and c = 5 b 4ac = ( 4) 4(1)(5) = 16 40 = 4 There are no real solutions. Martin-Gay, Developmental Mathematics 5
Solving Quadratic Equations Steps in Solving Quadratic Equations 1) If the equation is in the form (ax+b) = c, use the square root property to solve. ) If not solved in step 1, write the equation in standard form. 3) Try to solve by factoring. 4) If you haven t solved it yet, use the quadratic formula. Martin-Gay, Developmental Mathematics 53
Solving Equations Example Solve 1x = 4x + 4. 0 = 4x 1x + 4 0 = 4(x 3x + 1) Let a = 1, b = -3, c = 1 x 3 ( 3) (1) 4(1)(1) 3 9 4 3 5 Martin-Gay, Developmental Mathematics 54
Example Solving Equations Solve the following quadratic equation. 5 1 m m 8 5m 8m 4 0 0 ( 5m )( m ) 0 5m 0 or m m or m 5 0 Martin-Gay, Developmental Mathematics 55
/7/17 Complex Number Martin-Gay, Developmental Mathematics 56
What imaginary roots actually look like on a graph (which is why we don t usually graph them) Martin-Gay, Developmental Mathematics 57
Before, negative But now a special symbol is no solution. i is assigned so we can carry on the computation. 1 i Martin-Gay, Developmental Mathematics 58
Examples i 1 16 16 1 16 81 1 4 i 9 i 4i 9i Martin-Gay, Developmental Mathematics 59
i 1 Always! i 1 1 i i 1 ALWAYS! i 3 i i i i i 1i i 4 i iiii i i 1 1 1 i i 5 6 iiiii i i i i iiiiii i i i 1 the pattern repeats... Martin-Gay, Developmental Mathematics 60
Simplify i 6 Method Divide the exponent by 4 (or ) and look at the remainder. 1 6 4=6 with remainder Method i 6 i 1 i 13 13 6 =13 ( 1) 1 Martin-Gay, Developmental Mathematics 61
Simplify each expression. a] 8i 3i 4i 4 1 4 b] 5 0 i 5 i 0 Remember that 1 i i 100 110 10 Martin-Gay, Developmental Mathematics 6
When adding or subtracting complex numbers, combine like terms. Ex: 8 3i 5i 8 3i 5i 10 i Martin-Gay, Developmental Mathematics 63
Multiplication: 8 5i 3i F O I L 16 4i 10i 15i 16 14i 15 31 14i Martin-Gay, Developmental Mathematics 64
Complex Numbers Warm-Ups 8 minutes Multiply and Simplify : a} i 30 45 i b} (3 4 i) c} (3 4 i)(3 4 i) d} (4 i)(4 i) Martin-Gay, Developmental Mathematics 65
Division & Rationalizing the Denominator Ex] 5 3 5 3 3 5 3 3 3 Denominator can only be an integer! 1 ex] 36i 1 3i i i i 3i i i 3 3 Try] 18 3i 7i 6 i 6 i 9i 9i i i 6 16 9i 9 i i i Martin-Gay, Developmental Mathematics 66
Warm-up 8 minutes Simplify : a] i i 0 55 b] (4 5 i)(4 5 i) c] ( 6 )( 6 ) Solve using Quadratic Formula, what is the discriminant? d y x x ] Martin-Gay, Developmental Mathematics 67
Radical expression and Exponents Exponents v.s. Radicals (Roots) 5 3 15 therefore 3 15 5 Martin-Gay, Developmental Mathematics 68
Other examples 4 4 4 3 8 3 3 8 4 16 16 4 4 Martin-Gay, Developmental Mathematics 69
More Examples: a] 64 3 b] 64 c d e f 4 ] 64 5 ] 64 6 ] 64 7 ] 64 3 4 5 6 7 only 6 of them, not enough 3 8 4 7 4 5 4 64 Martin-Gay, Developmental Mathematics 70
More Examples: a] 81 3 b] 81 c d 4 ] 81 5 ] 81 3 4 5 3 3 3 3 3 3 3 3 3333 3333 Not enough numbers 3 9 3 3 3 3 still 3 81 Martin-Gay, Developmental Mathematics 71
3 8 8 8 8 51 1 3 3 8 8 3 4 16 16 16 16 16 65536 4 16 16 4 1 4 Try 6 79 on calculator Rational Exponents did you get 9? Rational Exponent is actually for calculators 16 ^ ( 1 4 ) Martin-Gay, Developmental Mathematics 7 x y
Practice NOW! DOK1 Simplify : a] ( 4 i )( 7 i ) b] ( 3 i )( 3 i ) c] ( 3 )( 3 ) d] i i 103 40 e] ( 8i ) DOK: a] f ( x) x 8x 3 b] g( x) x 6x 10 vertex d ] Solve? x int? y int? sketch c] Solve x 3 x ( x 4) 6 8 Martin-Gay, Developmental Mathematics 73
Answers a] 4 i b]13 c]1 d] i1 e] 60 3i vertex(, 5) 8 40 x int 4 0.4, 3.58 y int (0,3) vertex(3,1) 6 4 x int 6 i 3 y int (0,10) i Martin-Gay, Developmental Mathematics 74
Rationalization for Binomials When you have a Binomial for Denominator: a. 5i 3 i 3i 3i 64i15i 10i 96i6i4i 1611i 13 16 11 13 13 i b. 4 6 7 44 7 6 7 6 7 4 4 7 366 76 7 7 9 Martin-Gay, Developmental Mathematics 75
Ex: Solve x + 6x +10 = 0 x a = b = c = b b 4ac a 6 36 4110 1 6 6 4110 1 nd 1 st 6 36 40 6 4 6i 6 i 6 i and 3 i and 3i Martin-Gay, Developmental Mathematics 76