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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 91
Chapter 2, Problem 91

Solve each equation. (x-1)2/3+(x-1)1/3 -12 = 0

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1
Start by making a substitution to simplify the equation. Let \( y = (x - 1)^{1/3} \). This means \( y^2 = (x - 1)^{2/3} \).
Rewrite the original equation \( (x-1)^{2/3} + (x-1)^{1/3} - 12 = 0 \) in terms of \( y \) as \( y^2 + y - 12 = 0 \).
Recognize that the equation \( y^2 + y - 12 = 0 \) is a quadratic equation in standard form. Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a=1, b=1, c=-12 \) to find the values of \( y \).
After finding the values of \( y \), substitute back \( y = (x - 1)^{1/3} \) to get \( (x - 1)^{1/3} = y \).
Solve each resulting equation by cubing both sides to eliminate the cube root, giving \( x - 1 = y^3 \), and then solve for \( x \) by adding 1 to both sides.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Exponents

Rational exponents represent roots and powers simultaneously, where the numerator is the power and the denominator is the root. For example, x^(2/3) means the cube root of x squared. Understanding how to manipulate and simplify expressions with rational exponents is essential for solving the given equation.
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Substitution Method

The substitution method involves replacing a complex expression with a single variable to simplify the equation. In this problem, letting y = (x-1)^(1/3) transforms the equation into a quadratic form, making it easier to solve. After solving for y, substitute back to find x.
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Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two and can be solved by factoring, completing the square, or using the quadratic formula. Once the substitution reduces the original equation to a quadratic in y, these methods help find the roots, which then lead to the solutions for x.
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Related Practice
Textbook Question

Solve each equation. (2x-1)2/3+2(2x-1)1/3-3=0

Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | x2 + 5x + 5 | = 1

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Textbook Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9.

8x2−72=08x^2 - 72 = 0

Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these.

∣y∣=−x |y| = -x

Textbook Question

Solve each problem. A baseball is hit so that its height, s, in feet after t seconds is s=-16t2+44t+4. For what time period is the ball at least 32 ft above the ground?

Textbook Question

Simplify each power of i. i26

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