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Fields Medal Prize Winners (1998)




TUTORIALS:


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
Fractions
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
Roots
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
Powers
Rewriting Algebraic Fractions
Exponents
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 Quadratic Equations by Factoring

Solving an Equation of the Form ax2 + bx + c = 0 by Factoring

The number 0 has several special properties, including this one:

• The product of 0 and any number is 0. For example, 2 · 0 = 0.

Conversely, if the product of two real numbers is 0, then one or both must be 0. This statement is known as the Zero Product Property.

 

Property — Zero Product Property

English If the product of two numbers is 0, then one or both of the numbers is 0.

Algebra If a and b are real numbers, and if a · b = 0, then a = 0 or b = 0 or both a and b are equal to 0.

Example If (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0.

The Zero Product Property is useful for solving certain types of equations.

 

Example

Solve: (x - 3)(x + 5) = 0

Solution

The product of (x - 3) and (x + 5) is zero. (x - 3)(x + 5) = 0
Use the Zero Product Property to write two separate equations.

Solve each equation.

x - 3 = 0

x = 3

or

or

x + 5 = 0

x = -5

Thus, the original equation has two solutions: 3 and -5.

The solutions may be checked by substituting each value of x into the original equation and simplifying.

Check x = 3. Check x = -5.

(x - 3)(x + 5)

= 0

(x - 3)(x + 5)

= 0
Is

Is

Is

(3 - 3)(3 +5)

(0)(8)

0

= 0 ?

= 0 ?

= 0 ? Yes

Is

Is

Is

(-5 - 3)(-5 + 5)

(-8)(0)

0

= 0 ?

= 0 ?

= 0 ? Yes

 

Definition — Quadratic Equation

A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a 0.

A quadratic equation written in this form is said to be in standard form.

Now we will rewrite this equation.

We multiply the binomials.

We combine like terms.

(x - 3)(x + 5) = 0

x2 + 5x - 3x - 15 = 0

x2 + 2x - 15 = 0

This quadratic equation is written in standard form.

Notice that the terms on the left side of ax2 + bx + c = 0 are arranged in descending order by degree. The right side of the equation is zero.

Quadratic equations are also called second-degree equations because the degree of the polynomial, ax2 + bx + c, is 2.

We will use the Zero Product Property to solve some quadratic equations by factoring.

 

Procedure — To Solve a Quadratic Equation By Factoring

Step 1 Write the quadratic equation in the form ax2 + bx + c = 0.

Step 2 Factor the polynomial.

Step 3 Use the Zero Product Property.

Step 4 Solve each equation.

Step 5 Check each answer.