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Quadratic Equations
LEARNING OBJECTIVES
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.
Solve quadratic equations by using the quadratic formula.
Two televisions side-by-side. The right television is slightly larger than the left.
Figure 1.
The computer monitor on the left in (Figure) 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 as2x2+3x−1=0andx2−4=0are 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 ifa⋅b=0,thena=0orb=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(x−2)(x+3)by multiplying the two factors together.
(x−2)(x+3) = x2+3x−2x−6 = x2+x−6
The product is a quadratic expression. Set equal to zero,x2+x−6=0is 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,ax2+bx+c=0,where a, b, and c are real numbers, anda≠0.The equationx2+x−6=0is 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
If a⋅b=0, then a=0 or b=0,
where a and b are real numbers or algebraic expressions.
A quadratic equation is an equation containing a second-degree polynomial; for example
ax2+bx+c=0
where a, b, and c are real numbers, and ifa≠0,it is in standard form.
Solving Quadratics with a Leading Coefficient of 1
In the quadratic equationx2+x−6=0,the leading coefficient, or the coefficient ofx2,is 1. We have one method of factoring quadratic equations in this form.
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.