Solving equations

Solving polynomial equations by factoring:

Solve:

3X = 48X Write the equation in standard form (zero on the right side).

3X - 48X = 0 Take out the common factor and

3X(X - 16) = 0 collect the remaining factor.

3X(X - 4)(X + 4) = 0 Factor completely.

3X = 0 X - 4 = 0 X + 4 = 0 Set each factor equal to zero.

X = 0 X = 4 X = -4 Solve for X.

Solve:

X - 3X +2 = 0 Factor the left side of the equation as the product of 2 second-degree polynomials.

(X - 1)(X - 2) = 0 Partially factored

(X - 1)(X + 1)(X- 2) = 0 Completely factored

X - 1 = 0 X + 1 = 0 X - 2 = 0 Set each factor equal to zero.

X = 1 X = -1 X= 2 Solve for X.

Solve:

X - 2X - 3X = 0

X(X - 2X - 3) = 0 Take out the common factor.

X(X + 1)(X - 3) = 0 Factor completely.

X = 0 X + 1 = 0 X - 3 = 0 Set each factor equal to zero.

X = 0 X = -1 X = 3 Solve for X.

Solving equations involving rational exponents:

The first step is to isolate the rational expression. Second, determine the rational exponent and raise both sides of the equation to the reciprocal exponent. Simplify and check your answers.

Example:

4X- 8 = 0

4X = 8 Add 8 to both sides of the equation.

X = 2 Divide both sides by 4.

(X)= (2) Raise both sides to the 2/3 power(reciprocal power).

X = 2 OR

Check:

4X- 8 = 0

4(2)- 8 = 0 Replace X by 2

4(2) - 8 = 0 Simplify

8 - 8 = 0

0 = 0 The statement is true; therefore, the solution checks.

Solving Radical Equations:

The basic approach to solving radical equations is to get rid of the radical equation. Get rid of the square root by squaring each side of the equation. Get rid of the cube root by cubing each side of the equation, etc.. When changing radical equations and those equations involving rational exponents we often get extraneous solutions (solutions that do not fit the original equation). Therefore, a check of each tentative solution is required.

Example:

Solving Equations involving fractions:

To solve an equation involving fractions, multiply both sides of the equation by the least common denominator of each term in the equation. This procedure will clear the equation of fractions. Solve the following equations that contain fractions and check the tentative solutions:

Example:

multiply each term of the equation by the common denominator x(x+1)

Solving equations involving absolute values:

To solve an equation involving an absolute value, consider the fact that the expression inside the absolute value can be positive or negative. This consideration results in two separate equations, each of which must be solved.

Example:

Both x = 5 and x = -1 are solutions.