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Finding the Greatest Common Factor
Factoring Trinomials
Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
Adding Fractions
Subtracting Fractions
The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
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Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
Multiplying and Dividing Rational Numbers
Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
Multiplying and Dividing Signed Numbers
Solving Systems of Equations
Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
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Solving Quadratic Equations

Solving quadratic equations means to find when the branches crosses the x axis. Completing the square finds the max/min point (vertex) which can be helpful for some  types of questions, but other types of question ask about points that cross the axis.

In order to solve the problem:

1. there are several ways to start this process. In grade ten you learned two:

a. Factoring

i. Using simple or complex factoring techniques find the points and

solve for 0 = (x + a)(x + b)

b. Quadratic Equation

i. The plug and play solving method

c. The new method is to us completing the square to solve the equation for the roots

i. Complete the square so that it is in the form 0 = a (x + h) 2 + k

ii. arrange the equation so that the square is on one side and the constant is on the other.

iii. Solve for the variable using exponent laws (since it is always squared, both sides are square rooted)

iv. Isolate for the variable

Possible outcomes for roots of a Quadratic equation:

1. 2 real roots

a. notice the values of the coeffiecients

i. if the vertex is above the x axis AND a is negative, then 2 real


ii. If the vertex is below the x axis and the value of a is positive, then

there are two real roots

2. 1 real root

a. notice there is no vertical displacement. If the vertex is on the x axis there is only 1 real root

3. no real roots (therefore two complex roots)

a. notice the location of the vertex

i. the vertex is above the x axis and the parabola points up, there are no real roots (the graph never crosses the x axis)

ii. the vertex is below the x axis and the parabola points down, there are no real roots. (the graph never crosses the x axis)


Example 1

kx - 8 = 2x 2

What values does k have for two distinct real roots, one real root or no real roots.

Two real roots

-2x 2  + kx - 8 > 0

k 2 - 4(-2)(-8) > 0

k 2 > 64

k > 8; k < -8

one root

-2x 2  + kx - 8 = 0

k 2 - 4(-2)(-8) = 0

k 2 = 64

k = 8; k = -8

no roots

-2x 2  + kx - 8 < 0

k 2 - 4(-2)(-8) < 0

k 2 < 64

k < 8; k > -8

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