Fields Medal Prize Winners (1998)


Adding and Subtracting Monomials
Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Solving Nonlinear Equations by Factoring

A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers and a 0.

For example, the following are linear equations:

3x + 2(x - 5) = -2 2y - 3 = 5 + 3y

A nonlinear equation is an equation that cannot be written in the form ax + b = 0.

For example, the following are nonlinear equations:

x2 + 5x + 6 = 0


Recall that a linear function is a function that can be written as f(x) = ax + b or y = ax + b, where a 0.

The equation 3x + 2(x - 5) = -2 be rewritten as 5x - 8 = 0, which is in the linear form ax + b = 0.


Some nonlinear equations can be solved by factoring.


Procedure — To Solve a Nonlinear Equation by Factoring

Step 1 Write the equation in standard form.

Step 2 Factor.

Step 3 Use the Zero Product Property.

Step 4 Solve for the variable.



A polynomial is in standard form when the terms are arranged in descending order by degree on the left side of the equation and the right side of the equation is 0. For example,

4x3 - 8x2 - 5x + 10 = 0 is in standard form.

Zero Product Property: If P · Q = 0 then P = 0 or Q = 0. Or, P = 0 and Q = 0.


Example 1

Solve for x:

Step 1 Write the equation in standard form.
Multiply each side by the LCD, 2x.


Subtract 12 from both sides.

Step 2 Factor.

Step 3 Use the Zero Product Property.

Step 4 Solve for the variable.

x2 - 4x

x2 - 4x - 12

(x - 6)(x + 2)

x - 6 = 0  or

x = 6 or

x2 - 4x - 12

= 12

= 0

= 0

x + 2 = 0

x = -2

= 0

The equation has the same solutions as x2 - 4x - 12 = 0.

Thus, the solutions are x = 6 and x = -2.

The graph of the corresponding functions, and f(x) = x2 - 4x - 12 are shown. Even though the graphs are different they both cross the x-axis at x = -2 and x = 6, the solutions of the equations.


To factor x2 - 4x - 12, find two integers whose product is -12 and whose sum is -4. They are -6 and 2.