2.5 Quadratic Equations

Solving Quadratic Equations by Factoring

An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as $2x^2+3x-1=0$ 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.

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 $a\cdot b=0,$ 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. For example, expand the factored expression $\left(x-2\right)\left(x+3\right)$ by multiplying the two factors together.

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

Solving Quadratics with a Leading Coefficient of 1

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

How To

Given a quadratic equation with the leading coefficient of 1, factor it.

1. Find two numbers whose product equals c and whose sum equals b.
2. Use those numbers to write two factors of the form $\left(x+k\right)\text{or }\left(x-k\right),$ 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 $\mathrm{-2},$ the factors are $\left(x+1\right)\left(x-2\right).$
3. Solve using the zero-product property by setting each factor equal to zero and solving for the variable.

Example 1

Factoring and Solving a Quadratic with Leading Coefficient of 1

Factor and solve the equation: $x^2+x-6=0.$

Solution

To factor $x^2+x-6=0,$ we look for two numbers whose product equals $\mathrm{-6}$ and whose sum equals 1. Begin by looking at the possible factors of $\mathrm{-6.}$

$$\begin{array}{c}1\cdot (\mathrm{-6})\\ (\mathrm{-6})\cdot 1\\ 2\cdot (\mathrm{-3})\\ 3\cdot (\mathrm{-2})\end{array}$$

The last pair, $3\cdot \left(\mathrm{-2}\right)$ sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors.

$$\left(x-2\right)\left(x+3\right)=0$$

To solve this equation, we use the zero-product property. Set each factor equal to zero and solve.

$$\begin{array}{ccc}\hfill \left(x-2\right)\left(x+3\right)& =& 0\hfill \\ \hfill \left(x-2\right)& =& 0\hfill \\ \hfill x& =& 2\hfill \\ \hfill \left(x+3\right)& =& 0\hfill \\ \hfill x& =& \mathrm{-3}\hfill \end{array}$$

The two solutions are $2$ and $\mathrm{-3.}$ We can see how the solutions relate to the graph in Figure 2. The solutions are the x-intercepts of $y=x^2+x-6=0.$

Figure 2

Try It #1

Factor and solve the quadratic equation: $x^2-5x-6=0.$

Example 2

Solve the Quadratic Equation by Factoring

Solve the quadratic equation by factoring: $x^2+8x+15=0.$

Solution

Find two numbers whose product equals $15$ and whose sum equals $8.$ List the factors of $15.$

$$\begin{array}{c}1\cdot 15\hfill \\ 3\cdot 5\hfill \\ (\mathrm{-1})\cdot (\mathrm{-15})\hfill \\ (\mathrm{-3})\cdot (\mathrm{-5})\hfill \end{array}$$

The numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve.

$$\begin{array}{ccc}\hfill \left(x+3\right)\left(x+5\right)& =& 0\hfill \\ \hfill \left(x+3\right)& =& 0\hfill \\ \hfill x& =& \mathrm{-3}\hfill \\ \hfill \left(x+5\right)& =& 0\hfill \\ \hfill x& =& \mathrm{-5}\hfill \end{array}$$

The solutions are $\mathrm{-3}$ and $\mathrm{-5.}$

Try It #2

Solve the quadratic equation by factoring: $x^2-4x-21=0.$

Example 3

Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares

Solve the difference of squares equation using the zero-product property: $x^2-9=0.$

Solution

Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property.

$$\begin{array}{ccc}\hfill x^2-9& =& 0\hfill \\ \hfill \left(x-3\right)\left(x+3\right)& =& 0\hfill \\ \hfill \left(x-3\right)& =& 0\hfill \\ \hfill x& =& 3\hfill \\ \hfill \left(x+3\right)& =& 0\hfill \\ \hfill x& =& \mathrm{-3}\hfill \end{array}$$

The solutions are $3$ and $\mathrm{-3.}$

Try It #3

Solve by factoring: $x^2-25=0.$

Solving a Quadratic Equation by Factoring when the Leading Coefficient is not 1

When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let’s review the grouping procedures:

1. With the quadratic in standard form, $a{x}^2+bx+c=0,$ multiply $a\cdot c.$
2. Find two numbers whose product equals $ac$ and whose sum equals $b.$
3. Rewrite the equation replacing the $bx$ term with two terms using the numbers found in step 1 as coefficients of x.
4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
5. Factor out the expression in parentheses.
6. Set the expressions equal to zero and solve for the variable.

Example 4

Solving a Quadratic Equation Using Grouping

Use grouping to factor and solve the quadratic equation: $4x^2+15x+9=0.$

Solution

First, multiply $ac:4\left(9\right)=36.$ Then list the factors of $36.$

$$\begin{array}{l}1\cdot 36\hfill \\ 2\cdot 18\hfill \\ 3\cdot 12\hfill \\ 4\cdot 9\hfill \\ 6\cdot 6\hfill \end{array}$$

The only pair of factors that sums to $15$ is $3+12.$ Rewrite the equation replacing the b term, $15x,$ with two terms using 3 and 12 as coefficients of x. Factor the first two terms, and then factor the last two terms.

$$\begin{array}{ccc}\hfill 4x^2+3x+12x+9& =& 0\hfill \\ \hfill x(4x+3)+3(4x+3)& =& 0\hfill \\ \hfill (4x+3)(x+3)& =& 0\hfill \end{array}$$

Solve using the zero-product property.

$$\begin{array}{ccc}\hfill (4x+3)(x+3)& =& 0\hfill \\ \hfill (4x+3)& =& 0\hfill \\ \hfill x& =& -\frac{3}{4}\hfill \\ \hfill (x+3)& =& 0\hfill \\ \hfill x& =& -3\hfill \end{array}$$

The solutions are $-\frac{3}{4},$ and $\mathrm{-3.}$ See Figure 3.

Figure 3

Try It #4

Solve using factoring by grouping: $12x^2+11x+2=0.$

Example 5

Solving a Polynomial of Higher Degree by Factoring

Solve the equation by factoring: $\mathrm{-3}{x}^3-5x^2-2x=0.$

Solution

This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out $-x$ from all of the terms and then proceed with grouping.

$$\begin{array}{ccc}\hfill \mathrm{-3}{x}^3-5x^2-2x& =& 0\hfill \\ \hfill -x(3x^2+5x+2)& =& 0\hfill \end{array}$$

Use grouping on the expression in parentheses.

$$\begin{array}{ccc}\hfill -x(3x^2+3x+2x+2)& =& 0\hfill \\ \hfill -x[3x(x+1)+2(x+1)]& =& 0\hfill \\ \hfill -x(3x+2)(x+1)& =& 0\hfill \end{array}$$

Now, we use the zero-product property. Notice that we have three factors.

$$\begin{array}{ccc}\hfill -x& =& 0\hfill \\ \hfill x& =& 0\hfill \\ \hfill 3x+2& =& 0\hfill \\ \hfill x& =& -\frac{2}{3}\hfill \\ \hfill x+1& =& 0\hfill \\ \hfill x& =& \mathrm{-1}\hfill \end{array}$$

The solutions are $0,$ $-\frac{2}{3},$ and $\mathrm{-1.}$

Try It #5

Solve by factoring: $x^3+11x^2+10x=0.$

Using the Square Root Property

When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property, in which we isolate the $x^2$ term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the $x^2$ term so that the square root property can be used.

Completing the Square

Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation known as completing the square. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square.

We will use the example $x^2+4x+1=0$ to illustrate each step.

1. Given a quadratic equation that cannot be factored, and with $a=1,$ first add or subtract the constant term to the right side of the equal sign.

   $$x^2+4x=\mathrm{-1}$$

2. Multiply the b term by $\frac{1}{2}$ and square it.

   $$\begin{array}{ccc}\hfill \frac{1}{2}(4)& =& 2\hfill \\ \hfill 2^2& =& 4\hfill \end{array}$$

3. Add ${\left(\frac{1}{2}b\right)}^2$ to both sides of the equal sign and simplify the right side. We have

   $$\begin{array}{ccc}\hfill x^2+4x+4& =& -1+4\hfill \\ \hfill x^2+4x+4& =& 3\hfill \end{array}$$

4. The left side of the equation can now be factored as a perfect square.

   $$\begin{array}{ccc}\hfill x^2+4x+4& =& 3\hfill \\ \hfill {(x+2)}^2& =& 3\hfill \end{array}$$

5. Use the square root property and solve.

   $$\begin{array}{ccc}\hfill \sqrt{{(x+2)}^2}& =& \pm \sqrt{3}\hfill \\ \hfill x+2& =& \pm \sqrt{3}\hfill \\ \hfill x& =& \mathrm{-2}\pm \sqrt{3}\hfill \end{array}$$

6. The solutions are $\mathrm{-2}+\sqrt{3},$ and $\mathrm{-2}-\sqrt{3}.$

Using the Quadratic Formula

The fourth method of solving a quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.

We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by $\mathrm{-1}$ and obtain a positive a. Given $a{x}^2+bx+c=0,$ $a\ne 0,$ we will complete the square as follows:

1. First, move the constant term to the right side of the equal sign:

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

2. As we want the leading coefficient to equal 1, divide through by a:

   $$x^2+\frac{b}{a}x=-\frac{c}{a}$$

3. Then, find $\frac{1}{2}$ of the middle term, and add ${\left(\frac{1}{2}\frac{b}{a}\right)}^2=\frac{b^2}{4a^2}$ to both sides of the equal sign:

   $$x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac{c}{a}$$

4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:

   $${\left(x+\frac{b}{2a}\right)}^2=\frac{b^2-4ac}{4a^2}$$

5. Now, use the square root property, which gives

   $$\begin{array}{ccc}\hfill x+\frac{b}{2a}& =& \pm \sqrt{\frac{b^2-4ac}{4a^2}}\hfill \\ \hfill x+\frac{b}{2a}& =& \frac{\pm \sqrt{b^2-4ac}}{2a}\hfill \end{array}$$

6. Finally, add $-\frac{b}{2a}$ to both sides of the equation and combine the terms on the right side. Thus,

   $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

The Discriminant

The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant, or the expression under the radical, $b^2-4ac.$ The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. Table 1 relates the value of the discriminant to the solutions of a quadratic equation.

Using the Pythagorean Theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem. It is based on a right triangle, and states the relationship among the lengths of the sides as $a^2+b^2=c^2,$ where $a$ and $b$ refer to the legs of a right triangle adjacent to the $\mathrm{90°}$ angle, and $c$ refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.

We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.

The Pythagorean Theorem is given as

where $a$ and $b$ refer to the legs of a right triangle adjacent to the ${90}^{\circ }$ angle, and $c$ refers to the hypotenuse, as shown in Figure 4.

Figure 4

Example 12

Finding the Length of the Missing Side of a Right Triangle

Find the length of the missing side of the right triangle in Figure 5.

Figure 5

Solution

As we have measurements for side b and the hypotenuse, the missing side is a.

$$\begin{array}{ccc}\hfill a^2+b^2& =& c^2\hfill \\ a^2+{(4)}^2\hfill & =& {(12)}^2\hfill \\ \hfill a^2+16& =& 144\hfill \\ \hfill a^2& =& 128\hfill \\ \hfill a& =& \sqrt{128}\hfill \\ & =& 8\sqrt{2}\hfill \end{array}$$