OHM College Algebra — Interactive Demo

Quadratic Equations: Learn It 1

Learning Goals

Solve quadratic equations by factoring.
Solve quadratic equations by square root property.
Solve quadratic equations by completing the square.
Solve quadratic equations by using quadratic formula.

Quadratic Equations

An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as $2 x^{2} + 3 x - 1 = 0$ and $x^{2} - 4 = 0$ are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.

Key Takeaway

quadratic equations

A quadratic equation is a type of polynomial equation of the second degree, which means it involves at least one term that is squared (i.e., raised to the power of two).

The standard form of a quadratic equation is:

$a x^{2} + b x + c = 0$

where $a, b,$ and $c$ are real numbers and $a \neq = 0$ .

Solving Quadratic Equations by Factoring

Often the easiest method of solving a quadratic equation is factoring. Factoring a quadratic equation involves expressing it as a product of simpler polynomials, typically two binomials.

If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property which states that if $a \cdot b = 0$ , then $a = 0$ or $b = 0$ , where $a$ and $b$ are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.

Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring.

Key Takeaway

zero-product property

The zero-product property states

If $a \cdot b = 0,$ then $a = 0$ or $b = 0,$

where $a$ and $b$ are real numbers or algebraic expressions.

Example

For example, expand the factored expression $(x - 2) (x + 3)$ by multiplying the two factors together.

$(x - 2) (x + 3) = x^{2} + 3 x - 2 x - 6 = x^{2} + x - 6$

The product is a quadratic expression. Set equal to zero, $x^{2} + x - 6 = 0$ is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.

The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer.

Question Help

How To: Given a quadratic equation with the leading coefficient of 1, factor it

Find two numbers whose product equals $c$ and whose sum equals $b$ .
Use those numbers to write two factors of the form $(x + k)$ or $(x - k)$ , where $k$ is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are $1$ and $- 2$ , the factors are $(x + 1) (x - 2)$ .
Solve using the zero-product property by setting each factor equal to zero and solving for the variable.

Quadratic Equations: Learn It 1

Quadratic Equations

Solving Quadratic Equations by Factoring

Welcome to OHM