A.5 Methods for
Solving Equations
Failures are finger posts on the road to achievement.
-C. S. Lewis
Domain
Recall: The domain is the set of all x values for which an
expression exists.
Ex. Find the domain of the following expression.
3x  2
3x 3  4 x 2  9x 12

Solving Rational Equations
Recall how to solve the following equation…
x 1 1 5x5x
  
2 3 36 6

Steps for Solving Rational Equations
Step 1: Factor all denominators completely.
Step 2: Determine the least common multiple.
Step 3: Multiply both sides by the LCM (distribute!).
Step 4: Solve the new polynomial equation using learned
methods.
Step 5: Check for any extraneous solutions in the original
equation.
Solving Rational Equations
Solve the equation.
7
3
11
7



1 (x 1)(x  2) (x 1)(x  2)
x  2 xx 1

Solving Rational Equations
What if we have one a little more tricky?!
5
2
 2
3x 1 3x  5x  2


Solving Quadratic Equations: GCF
Solve the equation.
x 2  2x
Solve the equation.
x 3  6x 2

Completing the Square (a = 1)
x 2  6x  4  0
x 2  6x
 4
Step 1: Move c to the other side
Step 2:


x  6x  9  5
2
b 2
Add   to
2 
Step 
3: Rewrite the left side as
2
the perfect square  b 
x  
 2 
(x  3) 2  5

both sides
Step 4:
both sides and solve!



Completing the Square (a ≠ 1)
Solve the equation.
4 x 2  8x  1  0

Equations in Quadratic Form
The following equation resembles quadratic form. Let’s solve it!
2x 4  9x 2  35  0

Equations in Quadratic Form
How can we solve this equation without multiplying it out?
(3x  4)2  6(3x  4) 16  0

Solving Absolute Value Equations
What does “absolute value” mean?
x 7
What is the meaning of this equation?

2x  3  7
Steps for Solving Absolute Value Equations
Step 1: Isolate absolute value on one side of the equation.
x a
Step 2: Rewrite the equation, removing the absolute value,
and plus/minus of the other side of the equation.
x  a

Step 3: Solve each equation for x.
xa
x  a

Step 4: Check for any extraneous solutions in the original
equation.

Solving Absolute Value Equations
Solve the following equation for x.
3x  1  9  2x

Homework: p.998
#45 – 59 Odd
#77 – 89 Odd
#105 & 109
A.5 Methods for
Solving Equations
Failures are finger posts on the road to achievement.
-C. S. Lewis