Other Types of
Equations
Solving a Polynomial Equation
by Factoring
1. Move all terms to one side and obtain zero on
the other side.
2. Factor.
3. Apply the zero-product principle, setting each
factor equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.
Text Example
• Solve by factoring: 3x4 = 27x2.
Step 1 Move all terms to one side and
obtain zero on the other side. Subtract
27x2 from both sides
3x4 - x2 = 27x2 - 27x2
3x4 - 27x2 = 0
Step 2 Factor.
3x4 - 27x2 = 0
3x2(x2 - 9)= 0
Solution cont.
• Solve by factoring: 3x4 = 27x2.
Steps 3 and 4 Set each factor equal to
zero and solve each resulting equation.
3x2 = 0
or x2 - 9 = 0
x2 = 0
x2 = 9
x = 0
x = 9
x=0
x = 3
Steps 5 check your solution
Example
Solve:
Answer:
x -3 7 = 9
x-3 = 2
x-3 = 4
x=7
Radical Equations
A radical equation is an equation in which the
variable occurs in a square root, cube root or
higher root.
Example
Solve: 3  x  x - 5 = 8
Solution:
Answer:
•
Isolate the radical by moving the
other terms to the one side
x - 5 = 25 - 10 x  x 2
•
Square both sides to remove the
radical
0 = x 2 - 11x  30
•
Move all terms to one side
•
Factor
•
CHECK EACH “ANSWER”!!!!
Only one works!!!!
x -5 = 5- x
0 = ( x - 5)( x - 6)
x = 5,6
x=5
Solving Radical Equations of the
Form xm/n= k
•
Assume that m and n are positive integers,
m/n is in lowest terms, and k is a real
number.
1. Isolate the expression with the rational
exponent.
2. Raise both sides of the equation to the n/m
power.
Solving Radical Equations of the
Form xm/n= k cont.
If m is even:
If m is odd:
xm/n = k
xm/n = k
(xm/n) n/m = ±k
(xm/n)n/m = kn/m
x = ±kn/m
x = kn/m
It is incorrect to insert the ± when the numerator of
the exponent is odd. An odd index has only one
root.
3. Check all proposed solutions in the original
equation to find out if they are actual solutions
or extraneous solutions.
Text Example
Solve: x2/3 - 3/4 = -1/2.
Isolate x2/3 by adding 3/4 to both sides of the
equation: x2/3 = 1/4.
Raise both sides to the 3/2 power: (x2/3)3/2 =
±(1/4)3/2.
x = ±1/8.
Equations That Are Quadratic in Form
Some equations that are not quadratic can be
written as quadratic equations using an appropriate
substitution. Here are some examples.
Given Equation
Substitution New Equation
x4 – 8x2 – 9 = 0
or
(x2)2 – 8x2 – 9 = 0
t = x2
t2 – 8t – 9 = 0
5x2/3 + 11x1/3 + 2 = 0
or
5(x1/3)2 + 11x1/3 + 2 = 0
t = x1/3
5t2 + 11t + 2 = 0
Rewriting an Absolute Value Equation
without Absolute Value Bars
• If c is a positive real number and X
represents any algebraic expression, then
|X| = c is equivalent to X = c or X = -c.
Example
Solve:
3x - 1 = 4
Answer: 3x-1=4 and 3x-1=-4
solve,
3x=5
3x=-3
x=5/3
x=-1
Other Types of
Equations