## Solving Quadratic Equations Review

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AP Notes, Outlines, Study Guides, Vocabulary, Practice Exams and more!

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Section 4-6: The Quadratic Formula and the Discriminant The standard form of a quadratic is: +('' The quadratic formula is: )(. - - , where a, b, and c are real numbers where a # 0. - > What is the purpose of using the quadratic formula? : / What is the expression used to find the discriminant: \ > What information does the discriminant tell you? / (IC IS Value of Discriminant: ( Number and types of 'I \ ( solutions: Graph will look like: Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. Ex 1) a. x2 +6x+11 b.x2 +6x+9 c.x2 +6x+5 1 - - L - Identify the discrimant of-eacb of-the following equations and state how many solutions and what

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?6= ?2 b) 25 c) 7(x-4)2 ?18=10 tt fS )c 100 - f(TI1)jii Solving Quadratic Equations by Finding Square Roots Objective: To solve quadratic equations with real solutions. If R2=S, then Risa (00+ of S. A positive number S has 2 square roots written as ( and . ?O Properties: Product Property: V'-a- Quotient Property:Tb Simplify the expression (radical) a) b) c) id * 1-5 Rationalize Denominators of fractions 1 1 5D re: a, / 1,0 a) J ~S? I- b) 2 rOL - I Steps to solving quadratic equations: Step 1: Write the original equation - V+3f 1:-- ?4 } d) I? '- V64 '4L.f Form of the denominator Multiply numerator and denominator by: (multiply the _by_ _conjugate) puC 3 - j --3)-2 i_i0 (7+) ?- F (p

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Section 4.3: Solve by Factoring Vocabulary ? Monomial: OA 4?ct&in j?W ? Binomial: cuij e'ICS 1uX i-ewis Trinomial: CLAA Cc(.1 tJ9 :!2i *ermS *Some binomials and trinomials will have a Greatest Common Factor (GCF) that we will factor ou7 when factoring ? Solutions to a Quadratic Equation: X-'zef1 (O*5 so W )(-V 0 Greatest Common Factoring Ex1: Factor 3x2 +9x QCF? Ex 2: Factor 4x + 6 .. 3x(*-s x+3) Factoring a Trinomial with a leading coefficient of 1 (iejjx2 + bx + c) -Iind luy 'f&r&c -mc** Ex 3: Factor X2 - 9x+ 20 C, (x-5 I X-4 ) Ex 5: Factor x2 - 3x ?18 (-)(x+3) (14-C oxILt Ex4: Factor x2_3x+93 a tb :: -3 - 01 rj -.t - - - - Special Factoring Patterns ? Difference of Perfect Squares: a2 - - b)(a + b) Ex 6: Factor x2 - 9 xX 3

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Name: Date: Section 4-2 Notes Graphing Quadratic Functions in ird For Standard form of a quadratic function is E ax - A The parent function of the family of all quadratic functions is f(x) The graph of a quadratic function is a DaYWO01 IA. The vertex of a parabola is the or point on the parabola. .-.J 00 The axis of jcnY1LeA-rJ7 divides the parabola into mirror images and passes through the Graph the function y = CoWare to Graph the function y = (_--x)2. Compare it to -1 '/q 00 Thej X~'M The graph of a parabola is more vertically stretched than the parent function if 1 1k 1 > I The graph of a parabola is more vertically compressed than the parent function if The graph of a parabola is more horizontally stretched than the parent function if" I

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Quadratic Equations
This is what a "Standard" Quadratic Equation looks like:
•The letters a, b and c are coefficients (you know those values). They can have any value, except that a can't be 0.
•The letter "x" is the variable or unknown (you don't know it yet)
Here is an example of one:
The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).
It is also called an "Equation of Degree 2" (because of the "2" on the x)
More Examples of Quadratic Equations:
In this one a=2, b=5 and c=3
This one is a little more tricky:
•Where is a? In fact a=1, as we don't usually write "1x2"
•b = -3
•And where is c? Well, c=0, so is not shown.

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Laws of Exponents
(am)(an) = am+n
(ab)m = ambm
(am)n = amn
a0 = 1
(am)/(an) = am-n
a-m= 1/(am)
Quadratic Formula
In an equation like ax2 + bx + c = 0
You can solve for x using the Quadratic Formula:
Binomial Theorem
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 ...and so on...
Difference of Squares
a2 - b2 = (a - b)(a + b)
Rules of Zero
0/x = 0 where x is not equal to 0.
a0 = 1
0a = 0
a*0 = 0
a/0 is undefined (you can't do it)

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A "quadratic" is a polynomial that looks like "ax2 + bx + c", where "a", "b", and "c" are just numbers. For the easy case of factoring, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term. For instance: Factor x2 + 5x + 6. I need to find factors of 6 that add up to 5. Since 6 can be written as the product of 2 and 3, and since 2 + 3 = 5, then I'll use 2 and 3. I know from multiplying polynomials that this quadratic is formed from multiplying two factors of the form "(x + m)(x + n)", for some numbers m and n. So I'll draw my parentheses, with an "x" in the front of each: (x ??? ? )(x??? ? ) Then I'll write in the two numbers that I found above: (x + 2)(x + 3)

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an easy way to remember this quadratic equation to sing it with a pop goes the weezle theme song

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