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In this section you will:
• Solve quadratic equations by factoring.
• Solve quadratic equations by the square root property.
• Solve quadratic equations by completing the square.

The computer monitor on the left in [link] is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.

## Solving quadratic equations by factoring

An equation containing a second-degree polynomial is called a quadratic equation    . For example, equations such as $\text{\hspace{0.17em}}2{x}^{2}+3x-1=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{x}^{2}-4=0\text{\hspace{0.17em}}$ are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.

Often the easiest method of solving a quadratic equation is factoring . Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.

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 $\text{\hspace{0.17em}}a\cdot b=0,$ then $\text{\hspace{0.17em}}a=0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}b=0,$ 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. For example, expand the factored expression $\text{\hspace{0.17em}}\left(x-2\right)\left(x+3\right)\text{\hspace{0.17em}}$ by multiplying the two factors together.

$\begin{array}{ccc}\hfill \left(x-2\right)\left(x+3\right)& =& {x}^{2}+3x-2x-6\hfill \\ & =& {x}^{2}+x-6\hfill \end{array}$

The product is a quadratic expression. Set equal to zero, $\text{\hspace{0.17em}}{x}^{2}+x-6=0\text{\hspace{0.17em}}$ 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. We will look at both situations; but first, we want to confirm that the equation is written in standard form, $\text{\hspace{0.17em}}a{x}^{2}+bx+c=0,$ where a , b , and c are real numbers, and $\text{\hspace{0.17em}}a\ne 0.\text{\hspace{0.17em}}$ The equation $\text{\hspace{0.17em}}{x}^{2}+x-6=0\text{\hspace{0.17em}}$ is in standard form.

We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor    (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.

## The zero-product property and quadratic equations

The zero-product property    states

where a and b are real numbers or algebraic expressions.

A quadratic equation    is an equation containing a second-degree polynomial; for example

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

where a , b , and c are real numbers, and if $\text{\hspace{0.17em}}a\ne 0,$ it is in standard form.

In the quadratic equation $\text{\hspace{0.17em}}{x}^{2}+x-6=0,$ the leading coefficient, or the coefficient of $\text{\hspace{0.17em}}{x}^{2},$ is 1. We have one method of factoring quadratic equations in this form.

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