Section 1.6 - More Equations: Rational equations that reduce to linear or quadratic, Odd roots, Radicals, Rational exponents, Quadratic in form

Section Objectives

1. Determine the values of the variable that are restricted from a rational expression.
2. Solve rational equations that reduce to linear or quadratic.
3. Solve equations using odd roots.
4. Solve radical equations.
5. Solve equations involving rational exponents.
6. Solve equations that are quadratic in form.

Rational Equations

A rational expression is an algebraic expression in which whole number powers of the variable may appear in the numerator or denominator of a fraction.

For example, these are rational expressions:

$\displaystyle \frac{3x}{x+5},$

$\displaystyle \frac{x^2+x}{3x^2+1},$

$8x-7$

For now, we will be most interested in rational expressions that involve only first and second powers of the variable.

Restricted Values of the Variable

Recall that division by zero is not defined. Since rational expressions may contain variables in the denominator, we must be careful to avoid any values for the variable that would result in a zero denominator. We'll call these values restricted values.

For example...

The value $x=2$ is restricted from the rational expression $\displaystyle \frac{5x}{2x-4}$. Do you see why?

• [1] What value of $r$ is restricted from the expression $\displaystyle \frac{r-3}{3r-15}$? (Solution)

• [2] What value of $x$ is restricted from the expression $\displaystyle \frac{x-6}{x^2-12x+36}$? (Solution)

• [3] What value of $t$ is restricted from the expression $\displaystyle \frac{t^2+3t-18}{t^2+15t+54}$? (Solution)

Solving Rational Equations that Reduce to Linear or Quadratic

We will study lots of rational expressions and rational equations throughout the semester. Right now, we are only interested in rational equations that can be reduced to linear equations. We'll look at two special cases:

1. rational equations that are proportions and can be solved by cross multiplying, and
2. rational equations in which fractions can be cleared by multiplication.

Important point: A solution of a rational equation can never be a restricted value. Be sure to check for this!

Examples

• [4] Solve for $x$: $\qquad \displaystyle \frac{-6}{x} = \frac{-15}{x-3}$ (Solution)

• [5] Solve for $v$: $\qquad \displaystyle \frac{1}{v-5}=\frac{2}{3v-15}$ (Solution)

• [6] Solve for $w$: $\qquad \displaystyle \frac{w}{w-4} = \frac{4}{w-4}-\frac{4}{5}$ (Solution)

• [7] Solve for $u$: $\qquad \displaystyle \frac{u-1}{4u}+\frac{1}{6} = \frac{1}{u}$ (Solution)

• [8] Solve for $x$: $\qquad \displaystyle -\frac{5}{4x-16}+1=-\frac{2}{x-4}$ (Solution)

• [9] Solve for $w$: $\quad \displaystyle \frac{1}{w-4}+\frac{3}{w+2} = \frac{10}{w^2-2w-8}$ (Solution)

• [10] Solve for $v$: $\quad \displaystyle v + \frac{2}{v} = 8 -\frac{13}{v}$ (Solution)

• [11] Solve for $x$: $\quad \displaystyle 4 + \frac{3}{x-2} = \frac{3}{(x-1)(x-2)}$ (Solution)

• [12] Solve for $x$: $\quad \displaystyle \frac{x-4}{x-5}-1=\frac{x-2}{x-6}$ (Solution)

Examples Involving Odd Roots

• [13] Solve for $w$: $\quad w^3=-8$ (Solution)

• [14] Solve for $r$: $\quad r^3 = 25$ (Solution)

• [15] Solve for $y$: $\quad (y+8)^3-16=0$ (Solution)

Radical Equations

Radical equations are equations involving radicals and roots. A typical approach for solving certain kinds of radical equations is to isolate the radical and "undo" it by raising each side of the equation to the appropriate power. It is crucial that you check your answers in the original problem.

Keep in mind that radicals with even indicies can never be negative!

Examples with Radicals

• [16] Solve for $w$: $\quad \sqrt{w}=-6$ (Solution)

• [17] Solve for $u$: $\quad \sqrt{u} - 14=0$ (Solution)

• [18] Solve for $x$: $\quad -1+\sqrt{x-12}=7$ (Solution)

• [19] Solve for $r$: $\quad 2=\sqrt{3r-12}-1$ (Solution)

• [20] Solve for $u$: $\quad u = \sqrt{5u+14}$ (Solution)

• [21] Solve for $w$: $\quad \sqrt[4]{w}=-4$ (Solution)

• [22] Solve for $t$: $\quad \sqrt[3]{t}=2$ (Solution)

Equations with Rational Exponents

A rational exponent is an exponent that can be written as a fraction involving whole numbers. Rational (fractional) exponents denotes radicals, and expressions involving fractional exponents can be written in radical form. The key idea is that

$\displaystyle x^{n/m} = \sqrt[m]{x^n} = \left( \sqrt[m]{x} \right)^n$.

Be careful about making sure that these kinds of expressions are defined!

For example, $x^{1/2} = \sqrt{x}$ is only defined when $x \ge 0$. All other $x$-values are restricted.

On the other hand, $x^{1/3} = \sqrt[3]{x}$ is defined for all real $x$.

To solve an equation involving rational exponents, we might

1. rewrite the equation with radicals and solve as above, or
2. use "undoing" by raising each side to the appropriate $m/n$ power.

Examples with Rational Exponents

• [23] Solve for $w$: $\quad w^{1/4} = -5$ (Solution)

• [24] Solve for $r$: $\quad r^{1/4}=3$ (Solution)

• [25] Solve for $u$: $\quad (6u+4)^{1/3} + 3 = 7$ (Solution)

• [26] Solve for $x$: $\quad (3x+1)^{1/4}+5=3$ (Solution)

• [27] Solve for $z$: $\quad (z+2)^{3/2} = 3$ (Solution)

• [28] Solve for $x$: $\quad (x+7)^{2/5} = -1$ (Solution)

Examples that are Quadratic in Form

Some equations can be reduced to simpler equations by means of a substitution of variables. Each of these can be reduced to quadratic and an appropriate substitution.

• [29] Solve for $y$: $\quad (y^2-11)^2-10(y^2-11)+25=0$ (Solution)

• [30] Solve for $x$: $\quad x^4-37x^2+36=0$ (Solution)

• [31] Solve for $w$: $\quad 2w^{2/3}=3w^{1/3}+20$ (Solution)