Transcript MAT 108 Quadratic Equations

Quadratic Equations
By
Dr. Carol A. Marinas
Solving Equations
• In the previous section, we solved
LINEAR equations. This means that the
highest exponent is 1.
• In this section, we will be solving
equations with the highest exponent of
2. These are called Quadratic
Equations.
What are Quadratic Equations?
• They are in the form:
ax2 + bx + c = 0
• Methods of Solving Quadratic Equations
(Put into above form first)?
– By Factoring (sometimes works)
– By Quadratic Formula (always works)
By Factoring
• Put in Standard Form
• Factor the polynomial
• Set each factor to 0
• Solve for the variable
• Check answer in
original equation
• x2 - 3x = 4
x2 - 3x - 4 = 0
• (x - 4) (x + 1) = 0
• x - 4 = 0 or
x+1=0
• x = 4 or x = -1
• 42 - 3(4) = 4
(-1)2 - 3(-1) = 4
By Quadratic Formula
• This method works all the time.
• It is usually used when the
Factoring method fails.
• Put in standard form.
• Determine the values of a, b,
and c.
The Quadratic Formula
_______
-b +  b2 - 4ac
• x = -----------------------2a
Example:
• x2 + 9x = -8
• Standard form: x2 + 9x + 8 = 0
• We could do this by Factoring but let’s
use the Quadratic Formula.
• a = 1, b = 9 and c = 8
-9 +  92 - 4(1)(8)
-9 + 49
• x = ------------------------ = ----------2(1)
2
Example Continues
x
•
•
•
•
-9 + 49
-9 + 7
= -------------- = ------2
2
x = (-9 - 7)/2
or x = (-9 + 7)/2
x = -8 or
x = -1
x2 + 9x = -8
Check: (-8)2 + 9(-8) = -8
(-1)2 + 9(-1) = -8
REMEMBER --• If you have a “squared” term, it is a
“quadratic” equation.
• You must put the “quadratic equation” into
standard form first. This means that all terms
are on the left side and the right side is 0.
Not necessary but I would multiply through to
eliminate the fractions.
• Use either the “factoring” or “quadratic
formula” method to solve.
• Finally, check your answers in the original
equation.