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Power of a Power Property of Exponents
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Solving Equations with Radicals and Exponents
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Using Intercepts for Graphing Linear Equations
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Multiplying Fractions
Solving Linear Equations Containing Fractions
Evaluating Polynomials
Multiplication Property of Square and Cube  Roots
Writing a Fraction in Simplest Form
Square Roots
Inequalities
The Pythagorean Theorem
Factoring The Difference of 2 Squares
Solving Polynomial Equations
Roots and Powers
Writing Linear Equations in Standard Form
Solving Nonlinear Equations by Substitution
Straight Lines
The Square of a Binomial
Solving Equations
Adding and Subtracting Like Fractions
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Finding the Equation of an Inverse Function
Slope of a Line
Rules for Nonnegative Integral Exponents

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Solving Nonlinear Equations by Substitution

Example

Solve for x:

Solution = 18
Step 1 Write the equation in quadratic form.    

Subtract 18 from both sides.

= 0
Step 2 Use an appropriate “u” substitution.

Substitute u for

 

u2 + 3u - 18

 

= 0

Step 3 Solve the resulting equation.

Factor the new equation.

Use the Zero Product Property.

Solve each equation for u.

(u + 6)(u - 3)

u + 6 = 0 or u - 3

u = -6 or u

= 0

= 0

= 3

Step 4 Substitute the original expression for u. = 3
Step 5 Solve for the original variable.

Multiply both sides of each equation by x.

Write each equation in standard form.

x2 + 4 = -6x or x2 + 4

x2 + 6x + 4 = 0 or x2 - 3x + 4

= 3x

= 0

Neither equation factors. So, we will use the quadratic formula to solve each:

So, there are four solutions:

The equation written in standard form is

The graph of the corresponding function, is shown.

The graph crosses the x-axis at the two locations corresponding to the two real solutions of the equation:

Note that x is in a denominator and so it cannot equal 0. Therefore, the line x = 0 is a vertical asymptote.