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Graphing and Writing Linear Functions
SOLVING EQUATIONS INVOLVING RATIONAL EXPONENTS
Linear Equations and Graphing
Systems of Linear Equations
Solving Polynomial Equations
Matrix Equations and Solving Systems of Linear Equations
Introduction Part II and Solving Equations
Linear Algebra
Graphing Linear Inequalities
Using Augmented Matrices to Solve Systems of Linear Equations
Solving Linear Inequalities
Solution of the Equations
Linear Equations
Annotated Bibliography of Linear Algebra Books
Write Linear Equations in Standard Form
Graphing Linear Inequalities
Introduction to Linear Algebra for Engineers
Solving Quadratic Equations
THE HISTORY OF SOLVING QUADRATIC EQUATIONS
Systems of Linear Equations
Review for First Order Differential Equations
Systems of Nonlinear Equations & their solutions
LINEAR LEAST SQUARES FIT MAPPING METHOD FOR INFORMATION RETRIEVAL FROM NATURAL LANGUAGE TEXTS
Quadratic Equations
Syllabus for Differential Equations and Linear Alg
Linear Equations and Matrices
Solving Linear Equations
Slope-intercept form of the equation
Linear Equations
DETAILED SOLUTIONS AND CONCEPTS QUADRATIC EQUATIONS
Linear Equation Problems
Systems of Differential Equations
Linear Algebra Syllabus
Quadratic Equations and Problem Solving
LinearEquations
The Slope-Intercept Form of the Equation
Final Exam for Matrices and Linear Equations
Linear Equations
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Quadratic Equations and Problem Solving

Quadratic Equations and Problem Solving

We now focus on solving quadratic equations. Recall we learned how to solve for linear equations
(section 2-3); we will also solve quadratic equations graphically, numerically, and symbolically.

Note -> a quadratic equation can have zero, one, or two real solutions.

Quadratic Equation: A quadratic equation in one variable is an equation that can be written in the
form ax^2 + bx + c = 0 , where a, b, and c are real numbers with a ≠ 0.

Solving Quadratic Equations

4 basic symbolic strategies:

Factoring, Square Root Property, Completing the Square, Quadratic Formula
Factoring -> based on the zero-product property, if ab = 0 , then a = 0 or b = 0 or both.

Ex: x^2 - 4 = 0

Square Root Property -> Let k be a nonnegative number. Then the solutions to the equation

Ex: x^2 - 4 = 0 ->

Completing the Square -> useful when solving quadratic equations that do not factor easily!
Recall from 3-1 that the equation can be solved using x^2 + kx + (k / 2)^2 = (x + k / 2)^2 for k constant.
Ex:

Quadratic Formula -> solutions to the quadratic equation ax^2 + bx + c = 0 , where a ≠ 0, are

Ex:

Discriminant: In the quadratic formula, the quantity b^2 – 4ac is called the discriminant. The
discriminant provides the number of real solutions to a quadratic equation.

Note -> When a quadratic equation is solved graphically, the parabola intersects the x-axis 0, 1,
or 2 times. Where it crosses x-axis is (are) the x-intercepts!

• If b^2 – 4ac > 0, there are 2 real solutions
• If b^2 – 4ac = 0, there is 1 real solution
• If b^2 – 4ac < 0, there are no real solutions (these may be complex numbers)

GRAPHICALLY
Graph the quadratic equation and locate the x-intercepts.
Ex: x^2 - 4 = 0

NUMERICALLY
Make a table for quadratic equation y, Wherever y = 0 then these are “the zeros of y”, the solutions
(recall x-intercepts)!
Ex: x^2 - 4 = 0

Problem Solving

Exs: