Math 2 TOOLKITS. Solving Quadratics: Shortcuts to Try Before Using the Quadratic Formula. The Quadratic Formula has 4 Messages for You!

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1 Math 2 TOOLKITS Solving Quadratics: Shortcuts to Try Before Using the Quadratic Formula The Quadratic Formula has 4 Messages for You! The Discriminant Factoring When a=1 The Difference of Squares Perfect Square Trinomials NOTE: Each Math 2 teach may have used different formats. Perfect Square Trinomials may not be in the toolkit already. Jan 6 11:10 AM 1

Math 2 Toolkit Linear Term bx Quadratic Term ax 2 TOOLKIT: Solving Quadratics: Shortcuts to Try before using the Quadratic Formula Get the quadratic terms x 2 's on one side of the equation and

2 Math 2 Toolkit Linear Term bx Quadratic Term ax 2 TOOLKIT: Solving Quadratics: Shortcuts to Try before using the Quadratic Formula Get the quadratic terms x 2 's on one side of the equation and numbers on the other side. Solve for x by taking the square root of both sides of the equation. ax 2 = c Examples: 2x 2 9 = 0 or 12 = 4 + 4x 2 Write in standard form: ax 2 + bx = 0 Find the greatest common factor (always includes x as a factor) Use the zero product property (ZPP) to set each factor equal to zero and solve to x. Examples: 2x 2 4x = 0 or 12x = 4x 2 Constant Term c ax 2 + bx + c = 0 Write in standard form: Use trinomial factoring (factors of ac that add to b) and the zero product property (ZPP). Also, can a be factored out of all terms? Examples: x 2 4x 12 = 0 or 3x 2 12x 36 = 0 Check your answer: Graph the quadratic on the calculator and use the graph, the table, or 2nd Calc 2:zero. Oct 21 5:29 AM 2

3 TOOLKIT: The Quadratic Formula has 4 Messages for You! My b My b 2 4ac tells you: tells you: 2a The x coordinate the vertex & the line of symmetry. of Let's figure it out! My tells you: Solving my whole self tells you: The x intercept, roots, zeroes, solution. How far to go on each side of the line of symmetry to get to the roots. Mar 3 5:45 AM 3

TOOLKIT: The Discriminant The Quadratic Formula The Discriminant is just the part under the radical Teacher recommendation: When using the Quadratic Formula, calculate the discriminant first so that

4 TOOLKIT: The Discriminant The Quadratic Formula The Discriminant is just the part under the radical Teacher recommendation: When using the Quadratic Formula, calculate the discriminant first so that you know... When b 2 4ac > 0 there are 2 different real x intercepts/zeros/roots because the square root of positive number has two answers. When b 2 4ac = 0 there 1 real x intercept/zero/root because the square root of zero equals zero. When b 2 4ac < 0 there are no real x intercepts/zeros/roots because we cannot take the square root of a negative number. Oct 31 5:24 PM 4

5 TOOLKIT: Factoring When a=1 3p 2 2p 5 1. Find the factors of ac that add up to b. 3. Replace the linear term 4. Factor by Grouping ac = 3( 5) = 15 1, 15 3, 5 1, 15 3, 5 3p 2 3p +5p 5 3p(p 1) + 5(p 1) (p 1)(3p+5) Mar 16 9:57 AM 5

6 TOOLKIT: The Difference of Squares Difference = subtraction! x 2 4 is in standard form. What is the factored form? Repeat after me, "Find the factors of "ac" that add up to "b"." So, x 2 4 can be written 1x 2 + 0x 4 a=1, b=0, c= 4 Factors of "ac" = (1)( 4) = 4 That add up to "b" 1, 4 2, 2 1,4 Squares = perfect squares! Like 9, 16, 25, x 2, y 8... (x+2)(x 2) Learn to recognize the pattern of both the standard form and factored form! Same terms! Different signs! Practice x 2 49 x 4 1 9x 2 64 x 4 y (x+1)(x 1) (x+3)(x 3) (x 2 y+6)(x 2 y 6) (5x+2)(5x 2) Mar 10 11:05 AM 6

TOOLKIT: Perfect Square Trinomials A perfect square trinomial is a quadratic

A perfect square trinomial has the form: Where ax 2 and c are perfect squares

Shortcut to factor a perfect square trinomial: take the square root of the

More examples: Factored Form (x+1) 2 = (x+1)(x+1) = x 2 + 2x + 1 (x 2) 2 = (x

7 TOOLKIT: Perfect Square Trinomials A perfect square trinomial is a quadratic expression that factors into two identical binomials. A perfect square trinomial has the form: Where ax 2 and c are perfect squares and b equals (make sure that the linear term fits this pattern). Shortcut to factor a perfect square trinomial: take the square root of the quadratic and constant terms copy the sign of the linear term. More examples: Factored Form (x+1) 2 = (x+1)(x+1) = x 2 + 2x + 1 (x 2) 2 = (x 2)(x 2) = x 2 4x + 4 Standard x x + 25 = (x+5)(x+5) = (x+5) 2 Form 25x 2 30x + 9 = (5x 3)(5x 3) = (5x 3) 2 Standard Form Factored Form Jan 12 3:14 PM 7

8 Math 3 Tookit Set Up Aug 17 11:20 AM 8

Math 3 Table of Contents Category* Toolkit Title Page # Alg Symbolic, Number Line, & Interval Notations 1 Alg Solving Quadratic Inequalities 2 Alg Solving Complex Inequalities 3

9 Math 3 Table of Contents Category* Toolkit Title Page # Alg Symbolic, Number Line, & Interval Notations 1 Alg Solving Quadratic Inequalities 2 Alg Solving Complex Inequalities 3 If you are using your Math 2 Toolkit, then you'll start a NEW Table of Contents at the start of your unused pages. * Algebra, Statistics/Probability, Geometry I Aug 17 11:23 AM 9

10 Category* Toolkit Title Page # II Category* Toolkit Title Page # III Continue using the roman numerals for a total of four pages. Repeat the header. Category* Toolkit Title Page # IV 1 This page is for your first Toolkit entry. Next, use Arabic numbering for the next 20 or 30 pages. 2 3 Aug 17 11:27 AM 10

11 Attachments M3 211 Organizer.docx

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make