Solving Quadratic Equations
A quadratic equation is an equation of the form ,
ax2 + bx + c = 0 ,
where a, b, and c are real numbers , with a ≠ 0 . The condition, a ≠ 0 , ensures
that the
equation actually does have a x2-term.
Solving Quadratic Equations
1. Solve by factoring
2. Solve by completing the square
3. Solve by quadratic formula : 
Note: A number (say a) is a double root or root of
multiplicity 2 if x − a
appears as a factor twice in a given solution.
Example: (x − 4)(x − 4) = 0 implies that x = 4 is a double root.
A Special Quadratic Equation
For any real number k , the equation x2 = k is equivalent to
.
If k > 0, then x2 = k has 2 real solutions.
k < 0, then x2 = k has no real solution.
k = 0, then 0 is the only solution to x 2 = k .
Examples: Solve
Solutions:
Thus, the solution set for the equation x2 = 36 is {-6, 6}
.
Thus, the solution set for the equation
is 
.
Solving Quadratic Equations by Completing the Square
To complete the square of x 2 + kx , add
to both sides. That
is, add the
square of half the coefficient of x to both sides.
Note: When completing the square, the coefficient on the
x2- term MUST BE
EQUAL to 1.
Examples:
Solve the fol lowing quadratic equations using completing the square.
(a) x2 − 5x
(b) x2 − 5x + 3 = 0
Solutions:
(a) Complete the square for x2 − 5x .
(e) Solve x2 − 5x + 3 = 0.
Solution: Subtract 3 from both sides and add 
to
both sides.
Thus, the solution set is
Solving Quadratic Equations using the Quadratic Formula
Quadratic Formula
The solution to ax2 + bx + c = 0 , with a ≠ 0 , is given by the formula,
provided b2 − 4ac ≥ 0 .
If b2 − 4ac > 0 , then there exists has 2 real roots.
b2 − 4ac < 0 , then there exist no real roots.
b2 − 4ac = 0 , then there exists one root (double root).
Example: Solve the quadratic equation, x2 − 5x + 3 = 0, using the quadratic
formula.
Solution: Since a = 1, b = −5, and c = 3,
Thus, the solution set is
.
Solving Quadratic Equations Assessment Quiz
Solve the following quadratic equations using the method indicated and choose
the correct answer.
1. Solve x2 + 2x −8 = 0 by factoring.
2. Solve x2 −10x + 5 = 0 by completing the square.
3 Solve x2 + 8x + 6 = 0 using the quadratic formula.
Solving Quadratic Equations Lesson Assessment
Solve each quadratic function using the method indicated. Please show all your
work.
1. Solve x2 − x − 20 = 0 by factoring.
2. Solve x2 + 6x +1 = 0 by completing the square.
3. Solve 2x2 −5x − 3 = 0 using the quadratic formula.
Graphing Piecewise Functions Lesson Module
Title of Unit
Graphing Piecewise Functions
Grade Level
Postsecondary
Learning Goal
At the end of this unit, the learner will be able to:
• Define a piecewise function.
• Recognize piecewise functions.
• Sketch the graph of a piecewise function.
• Recognize a graph of a piecewise function.
Learning Outcome(s)
Problem-solving
Objective(s)
Given access to a computer, the Internet, and Adam, the learner will be able to
construct a graphical inter pretation of a piecewise function.
