## Solving Quadratic Equations Review

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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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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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The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations.

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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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