Learning Outcomes
- Solve a radical equation, identify extraneous solution.
- Solve an equation with rational exponents.
- Solve polynomial equations.
- Solve absolute value equations.
We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.
Equations With Radicals and Rational Exponents
Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as
[latex]\begin{array}{ccc} \sqrt{3x+18}=x & \\ \sqrt{x+3}=x-3 & \\ \sqrt{x+5}-\sqrt{x - 3}=2\end{array}[/latex]
Radical equations may have one or more radical terms and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. Checking each answer in the original equation will confirm the true solutions.
A General Note: Radical Equations
An equation containing terms with a variable in the radicand is called a radical equation.
How To: Given a radical equation, solve it
- Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.
- If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an nth root radical, raise both sides to the nth power. Doing so eliminates the radical symbol.
- Solve the resulting equation.
- If a radical term still remains, repeat steps 1–2.
- Check solutions by substituting them into the original equation.
Example: Solving an Equation with One Radical
Solve [latex]\sqrt{15 - 2x}=x[/latex].
Show Solution
The radical is already isolated on the left side of the equal sign, so proceed to square both sides.
[latex]\begin{array}{lll}\sqrt{15 - 2x}=x & \\ {\left(\sqrt{15 - 2x}\right)}^{2}={\left(x\right)}^{2} & \\ 15 - 2x={x}^{2}\end{array}[/latex]
We see that the remaining equation is a quadratic. Set it equal to zero and solve.
[latex]\begin{array}{llll}0={x}^{2}+2x - 15 & \\ 0=\left(x+5\right)\left(x - 3\right) & \\ x=-5 & \\ x=3 \end{array}[/latex]
The proposed solutions are [latex]x=-5[/latex] and [latex]x=3[/latex]. Let us check each solution back in the original equation. First, check [latex]x=-5[/latex].
[latex]\begin{array}{llll}\sqrt{15 - 2x}=x & \\ \sqrt{15 - 2\left(-5\right)}=-5 & \\ \sqrt{25}=-5 & \\ 5\ne -5\end{array}[/latex]
This is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not satisfy the original equation.
Check [latex]x=3[/latex].
[latex]\begin{array}{llll}\sqrt{15 - 2x}=x & \\ \sqrt{15 - 2\left(3\right)}=3 & \\ \sqrt{9}=3 & \\ 3=3\end{array}[/latex]
The solution is [latex]x=3[/latex].
Try It
Solve the radical equation: [latex]\sqrt{x+3}=3x - 1[/latex]
Show Solution
[latex]x=1[/latex]; extraneous solution [latex]x=-\frac{2}{9}[/latex]
Example: Solving a Radical Equation Containing Two Radicals
Solve [latex]\sqrt{2x+3}+\sqrt{x - 2}=4[/latex].
Show Solution
As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.
[latex]\begin{array}{llllll}\sqrt{2x+3}+\sqrt{x - 2}=4\hfill & \hfill & \\ \sqrt{2x+3}=4-\sqrt{x - 2}\hfill & \text{Subtract }\sqrt{x - 2}\text{ from both sides}.\hfill & \\ {\left(\sqrt{2x+3}\right)}^{2}={\left(4-\sqrt{x - 2}\right)}^{2}\hfill & \text{Square both sides}.\hfill \end{array}[/latex]
Use the perfect square formula to expand the right side: [latex]{\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}[/latex].
[latex]\begin{array}{lllllllllll}2x+3={\left(4\right)}^{2}-2\left(4\right)\sqrt{x - 2}+{\left(\sqrt{x - 2}\right)}^{2}\hfill & \hfill & \\ 2x+3=16 - 8\sqrt{x - 2}+\left(x - 2\right)\hfill & \hfill & \\ 2x+3=14+x - 8\sqrt{x - 2}\hfill & \text{Combine like terms}.\hfill & \\ x - 11=-8\sqrt{x - 2}\hfill & \text{Isolate the second radical}.\hfill & \\ {\left(x - 11\right)}^{2}={\left(-8\sqrt{x - 2}\right)}^{2}\hfill & \text{Square both sides}.\hfill & \\ {x}^{2}-22x+121=64\left(x - 2\right)\hfill & \hfill \end{array}[/latex]
Now that both radicals have been eliminated, set the quadratic equal to zero and solve.
[latex]\begin{array}{llllllllll}{x}^{2}-22x+121=64x - 128\hfill & \hfill & \\ {x}^{2}-86x+249=0\hfill & \hfill & \\ \left(x - 3\right)\left(x - 83\right)=0\hfill & \text{Factor and solve}.\hfill & \\ x=3\hfill & \hfill & \\ x=83\hfill & \hfill \end{array}[/latex]
The proposed solutions are [latex]x=3[/latex] and [latex]x=83[/latex]. Check each solution in the original equation.
[latex]\begin{array}{lllll}\sqrt{2x+3}+\sqrt{x - 2}=4\hfill & \\ \sqrt{2x+3}=4-\sqrt{x - 2}\hfill & \\ \sqrt{2\left(3\right)+3}=4-\sqrt{\left(3\right)-2}\hfill & \\ \sqrt{9}=4-\sqrt{1}\hfill \\ 3=3\hfill \end{array}[/latex]
One solution is [latex]x=3[/latex].
Check [latex]x=83[/latex].
[latex]\begin{array}{lllll}\sqrt{2x+3}+\sqrt{x - 2}=4\hfill & \\ \sqrt{2x+3}=4-\sqrt{x - 2}\hfill & \\ \sqrt{2\left(83\right)+3}=4-\sqrt{\left(83 - 2\right)}\hfill & \\ \sqrt{169}=4-\sqrt{81}\hfill & \\ 13\ne -5\hfill \end{array}[/latex]
The only solution is [latex]x=3[/latex]. We see that [latex]x=83[/latex] is an extraneous solution.
Try It
Solve the equation with two radicals: [latex]\sqrt{3x+7}+\sqrt{x+2}=1[/latex].
Show Solution
[latex]x=-2[/latex]; extraneous solution [latex]x=-1[/latex]
Solving Equations With Rational Exponents
Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, [latex]{16}^{\frac{1}{2}}[/latex] is another way of writing [latex]\sqrt{16}[/latex]; [latex]{8}^{\frac{1}{3}}[/latex] is another way of writing [latex]\text{ }\sqrt[3]{8}[/latex]. The ability to work with rational exponents is a useful skill as it is highly applicable in calculus.
We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example, [latex]\frac{2}{3}\left(\frac{3}{2}\right)=1[/latex], [latex]3\left(\frac{1}{3}\right)=1[/latex], and so on.
A General Note: Rational Exponents
A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:
[latex]{a}^{\frac{m}{n}}={\left({a}^{\frac{1}{n}}\right)}^{m}={\left({a}^{m}\right)}^{\frac{1}{n}}=\sqrt[n]{{a}^{m}}={\left(\sqrt[n]{a}\right)}^{m}[/latex]
Example: Evaluating a Number Raised to a Rational Exponent
Evaluate [latex]{8}^{\frac{2}{3}}[/latex].
Show Solution
Whether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite [latex]{8}^{\frac{2}{3}}[/latex] as [latex]{\left({8}^{\frac{1}{3}}\right)}^{2}[/latex].
[latex]\begin{array}{l}{\left({8}^{\frac{1}{3}}\right)}^{2}\hfill&={\left(2\right)}^{2}\hfill \\ \hfill&=4\hfill \end{array}[/latex]
Try It
Evaluate [latex]{64}^{-\frac{1}{3}}[/latex].
Show Solution
[latex]\frac{1}{4}[/latex]
Example: Solve the Equation Including a Variable Raised to a Rational Exponent
Solve the equation in which a variable is raised to a rational exponent: [latex]{x}^{\frac{5}{4}}=32[/latex].
Show Solution
The way to remove the exponent on x is by raising both sides of the equation to a power that is the reciprocal of [latex]\frac{5}{4}[/latex], which is [latex]\frac{4}{5}[/latex].
[latex]\begin{array}{llllllll}{x}^{\frac{5}{4}}=32\hfill & \hfill & \\ {\left({x}^{\frac{5}{4}}\right)}^{\frac{4}{5}}={\left(32\right)}^{\frac{4}{5}}\hfill & \hfill & \\ x={\left(2\right)}^{4}\hfill & \text{The fifth root of 32 is 2}.\hfill & \\ x=16\hfill & \hfill \end{array}[/latex]
Try It
Solve the equation [latex]{x}^{\frac{3}{2}}=125[/latex].
Example: Solving an Equation Involving Rational Exponents and Factoring
Solve [latex]3{x}^{\frac{3}{4}}={x}^{\frac{1}{2}}[/latex].
Show Solution
This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.
[latex]\begin{array}{ll}3{x}^{\frac{3}{4}}-\left({x}^{\frac{1}{2}}\right)={x}^{\frac{1}{2}}-\left({x}^{\frac{1}{2}}\right)\hfill & \\ 3{x}^{\frac{3}{4}}-{x}^{\frac{1}{2}}=0\hfill \end{array}[/latex]
Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite [latex]{x}^{\frac{1}{2}}[/latex] as [latex]{x}^{\frac{2}{4}}[/latex]. Then, factor out [latex]{x}^{\frac{2}{4}}[/latex] from both terms on the left.
[latex]\begin{array}{ll}3{x}^{\frac{3}{4}}-{x}^{\frac{2}{4}}=0\hfill & \\ {x}^{\frac{2}{4}}\left(3{x}^{\frac{1}{4}}-1\right)=0\hfill \end{array}[/latex]
Where did [latex]{x}^{\frac{1}{4}}[/latex] come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply [latex]{x}^{\frac{2}{4}}[/latex] back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to [latex]\frac{2}{4}[/latex] equals [latex]\frac{3}{4}[/latex]. Thus, the exponent on x in the parentheses is [latex]\frac{1}{4}[/latex].
Let us continue. Now we have two factors and can use the zero factor theorem.
[latex]\begin{array}{llllllllllllll}{x}^{\frac{2}{4}}\left(3{x}^{\frac{1}{4}}-1\right)=0\hfill & \hfill & \\ {x}^{\frac{2}{4}}=0\hfill & \hfill & \\ x=0\hfill & \hfill & \\ 3{x}^{\frac{1}{4}}-1=0\hfill & \hfill & \\ 3{x}^{\frac{1}{4}}=1\hfill & \hfill & \\ {x}^{\frac{1}{4}}=\frac{1}{3}\hfill & \text{Divide both sides by 3}.\hfill & \\ {\left({x}^{\frac{1}{4}}\right)}^{4}={\left(\frac{1}{3}\right)}^{4}\hfill & \text{Raise both sides to the reciprocal of }\frac{1}{4}.\hfill & \\ x=\frac{1}{81}\hfill & \hfill \end{array}[/latex]
The two solutions are [latex]x=0[/latex], [latex]x=\frac{1}{81}[/latex].
Try It
Solve: [latex]{\left(x+5\right)}^{\frac{3}{2}}=8[/latex].
Solving Other Types of Equations
We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.
A General Note: Polynomial Equations
A polynomial of degree n is an expression of the type
[latex]{a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\cdot \cdot \cdot +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
where n is a positive integer and [latex]{a}_{n},\dots ,{a}_{0}[/latex] are real numbers and [latex]{a}_{n}\ne 0[/latex].
Setting the polynomial equal to zero gives a polynomial equation. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent n.
Solving an Absolute Value Equation
Next, we will learn how to solve an absolute value equation. To solve an equation such as [latex]|2x - 6|=8[/latex], notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is [latex]8[/latex] or [latex]-8[/latex]. This leads to two different equations we can solve independently.
[latex]\begin{array}{lll}2x - 6=8\hfill & \text{ or }\hfill & 2x - 6=-8\hfill \\ 2x=14\hfill & \hfill & 2x=-2\hfill \\ x=7\hfill & \hfill & x=-1\hfill \end{array}[/latex]
Knowing how to solve problems involving absolute value is useful. For example, we may need to identify numbers or points on a line that are a specified distance from a given reference point.
A General Note: Absolute Value Equations
The absolute value of x is written as [latex]|x|[/latex]. It has the following properties:
[latex]\begin{array}{l}\text{If } x\ge 0,\text{ then }|x|=x.\hfill \\ \text{If }x<0,\text{ then }|x|=-x.\hfill \end{array}[/latex]
For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex], will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]. If [latex]B<0[/latex], the equation [latex]|A|=B[/latex] has no solution.
An absolute value equation in the form [latex]|ax+b|=c[/latex] has the following properties:
[latex]\begin{array}{l}\text{If }c<0,|ax+b|=c\text{ has no solution}.\hfill \\ \text{If }c=0,|ax+b|=c\text{ has one solution}.\hfill \\ \text{If }c>0,|ax+b|=c\text{ has two solutions}.\hfill \end{array}[/latex]
How To: Given an absolute value equation, solve it
- Isolate the absolute value expression on one side of the equal sign.
- If [latex]c>0[/latex], write and solve two equations: [latex]ax+b=c[/latex] and [latex]ax+b=-c[/latex].
Example: Solving Absolute Value Equations
Solve the following absolute value equations:
- [latex]|6x+4|=8[/latex]
- [latex]|3x+4|=-9[/latex]
- [latex]|3x - 5|-4=6[/latex]
- [latex]|-5x+10|=0[/latex]
Show Solution
a. [latex]|6x+4|=8[/latex]
Write two equations and solve each:
[latex]\begin{array}{lllllllll}6x+4=8\hfill & \text{ or } & 6x+4=-8\hfill & \\ 6x=4\hfill & \hfill & 6x=-12\hfill & \\ x=\frac{2}{3}\hfill& \hfill & x=-2\hfill \end{array}[/latex]
The two solutions are [latex]x=\frac{2}{3}[/latex], [latex]x=-2[/latex].
b. [latex]|3x+4|=-9[/latex]
There is no solution as an absolute value cannot be negative.
c. [latex]|3x - 5|-4=6[/latex]
Isolate the absolute value expression and then write two equations.
[latex]\begin{array}{lll}\hfill & |3x - 5|-4=6\hfill & \hfill \\ \hfill & |3x - 5|=10\hfill & \hfill \\ \hfill & \hfill & \hfill \\ 3x - 5=10\hfill & \hfill & 3x - 5=-10\hfill \\ 3x=15\hfill & \hfill & 3x=-5\hfill \\ x=5\hfill & \hfill & x=-\frac{5}{3}\hfill \end{array}[/latex]
There are two solutions: [latex]x=5[/latex], [latex]x=-\frac{5}{3}[/latex].
d. [latex]|-5x+10|=0[/latex]
The equation is set equal to zero, so we have to write only one equation.
[latex]\begin{array}{ccc}-5x+10=0\hfill & \\ -5x=-10\hfill & \\ x=2\hfill \end{array}[/latex]
There is one solution: [latex]x=2[/latex].
Try It
Solve the absolute value equation: [latex]|1 - 4x|+8=13[/latex].
Show Solution
[latex]x=-1[/latex], [latex]x=\frac{3}{2}[/latex]
Other Types of Equations
There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form and rational equations that result in a quadratic.
Solving Equations in Quadratic Form
Equations in quadratic form are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include [latex]{x}^{4}-5{x}^{2}+4=0,{x}^{6}+7{x}^{3}-8=0[/latex], and [latex]{x}^{\frac{2}{3}}+4{x}^{\frac{1}{3}}+2=0[/latex]. In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.
A General Note: Quadratic Form
If the exponent on the middle term is one-half of the exponent on the leading term, we have an equation in quadratic form which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.
How To: Given an equation quadratic in form, solve it
- Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.
- If it is, substitute a variable, such as u, for the variable portion of the middle term.
- Rewrite the equation so that it takes on the standard form of a quadratic.
- Solve using one of the usual methods for solving a quadratic.
- Replace the substitution variable with the original term.
- Solve the remaining equation.
Example: Solving a Fourth-Degree Equation in Quadratic Form
Solve this fourth-degree equation: [latex]3{x}^{4}-2{x}^{2}-1=0[/latex].
Show Solution
This equation fits the main criteria: that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let [latex]u={x}^{2}[/latex]. Rewrite the equation in u.
[latex]3{u}^{2}-2u - 1=0[/latex]
Now solve the quadratic.
[latex]\begin{array}{ll}3{u}^{2}-2u - 1=0\hfill & \\ \left(3u+1\right)\left(u - 1\right)=0\hfill \end{array}[/latex]
Solve for u.
[latex]\begin{array}{lllll}3u+1=0\hfill & \\ 3u=-1\hfill & \\ u=-\frac{1}{3}\hfill & \\ {x}^{2}=-\frac{1}{3}\hfill & \\ x=\pm i\sqrt{\frac{1}{3}}\hfill \end{array}[/latex]
Replace u with its original term.
[latex]\begin{array}{llll}u - 1=0\hfill & \\ u=1\hfill & \\ {x}^{2}=1\hfill & \\ x=\pm 1\hfill \end{array}[/latex]
The solutions are [latex]x=\pm i\sqrt{\frac{1}{3}}[/latex] and [latex]x=\pm 1[/latex].
Try It
Solve using substitution: [latex]{x}^{4}-8{x}^{2}-9=0[/latex].
Show Solution
[latex]x=-3,3,-i,i[/latex]
Example: Solving an Equation in Quadratic Form Containing a Binomial
Solve the equation in quadratic form: [latex]{\left(x+2\right)}^{2}+11\left(x+2\right)-12=0[/latex].
Show Solution
This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution letting [latex]u=x+2[/latex]. Then rewrite the equation in u.
[latex]\begin{array}{ll}{u}^{2}+11u - 12=0\hfill & \\ \left(u+12\right)\left(u - 1\right)=0\hfill \end{array}[/latex]
Solve using the zero-factor property and then replace u with the original expression.
[latex]\begin{array}{llll}u+12=0\hfill & \\ u=-12\hfill & \\ x+2=-12\hfill & \\ x=-14\hfill \end{array}[/latex]
The second factor results in
[latex]\begin{array}{llll}u - 1=0\hfill & \\ u=1\hfill & \\ x+2=1\hfill & \\ x=-1\hfill \end{array}[/latex]
We have two solutions: [latex]x=-14[/latex], [latex]x=-1[/latex].
Try It
Solve: [latex]{\left(x - 5\right)}^{2}-4\left(x - 5\right)-21=0[/latex].
Solving Rational Equations Resulting in a Quadratic
Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.
Example: Solving a Rational Equation Leading to a Quadratic
Solve the following rational equation: [latex]\frac{-4x}{x - 1}+\frac{4}{x+1}=\frac{-8}{{x}^{2}-1}[/latex].
Show Solution
We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However, [latex]{x}^{2}-1=\left(x+1\right)\left(x - 1\right)[/latex]. Then, the LCD is [latex]\left(x+1\right)\left(x - 1\right)[/latex]. Next, we multiply the whole equation by the LCD.
[latex]\begin{array}{cccccccc}\left(x+1\right)\left(x - 1\right)\left[\frac{-4x}{x - 1}+\frac{4}{x+1}\right]=\left[\frac{-8}{\left(x+1\right)\left(x - 1\right)}\right]\left(x+1\right)\left(x - 1\right) \\ -4x\left(x+1\right)+4\left(x - 1\right)=-8 \\ -4{x}^{2}-4x+4x - 4=-8 \\ -4{x}^{2}+4=0 \\ -4\left({x}^{2}-1\right)=0 \\ -4\left(x+1\right)\left(x - 1\right)=0 \\ x=-1 \\ x=1 \end{array}[/latex]
In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.
Try It
Solve [latex]\frac{3x+2}{x - 2}+\frac{1}{x}=\frac{-2}{{x}^{2}-2x}[/latex].
Show Solution
[latex]x=-1[/latex], [latex]x=0[/latex] is not a solution.
Key Concepts
- Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve a radical equation, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1.
- Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping.
- We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
- To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value.
- Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve.
- Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form.
Glossary
- absolute value equation
- an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression
- equations in quadratic form
- equations with a power other than 2 but with a middle term with an exponent that is one-half the exponent of the leading term
- extraneous solutions
- any solutions obtained that are not valid in the original equation
- polynomial equation
- an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents
- radical equation
- an equation containing at least one radical term where the variable is part of the radicand