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 zeroproduct 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 Step 2 Factor.

3x4 27x2 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 = 0x2 = 0 x2 = 9x = 0 x = 9x = 0 x = 3

Steps 5 check your solution

Example

Solve:

Answer:

973 x

7

43

23

x

x

x

Radical Equations

A radical equation is an equation in which the variable occurs in a square root, cube root or higher root.

ExampleSolve:

Answer:

853 xx

5

6,5

)6)(5(0

30110

10255

55

2

2

x

x

xx

xx

xxx

xx

Solution:• Isolate the radical by moving the

other terms to the one side

• Square both sides to remove the radical

• Move all terms to one side

• Factor

• CHECK EACH “ANSWER”!!!! Only one works!!!!

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.

Some equations that are not quadratic can be written as quadratic equations using an appropriate substitution. Here are some examples.

5t2 + 11t + 2 = 0t = x1/35x2/3 + 11x1/3 + 2 = 0

or

5(x1/3)2 + 11x1/3 + 2 = 0

t2 – 8t – 9 = 0t = x2x4 – 8x2 – 9 = 0

or

(x2)2 – 8x2 – 9 = 0

New EquationSubstitutionGiven Equation

Equations That Are Quadratic in Form

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:

Answer: 3x-1=4 and 3x-1=-4

solve, 3x=5 3x=-3

x=5/3 x=-1

413 x

Other Types of Equations