Problem 37 - College Algebra

Question

Solve and check each equation with rational exponents. $$ \left(x^{2}-x-4\right)^{3 / 4}-2=6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Isolate the expression with the rational exponent

First, add 2 to both sides of the equation to get \(\left(x^{2}-x-4\right)^{3 / 4} = 8\)

Step 2: Remove the rational exponent

To do away with the exponent \(3/4\), raise both sides of the equation to the power \(4/3\). This results in \(x^{2} - x - 4 = 8^{4/3}\). After further simplification, \(x^2 - x - 4 = 16\).

Step 3: Simplify the equation

Rearrange the equation to form a quadratic equation \(x^2 - x - 20 = 0\).

Step 4: Factor the equation

Now, factor the equation. This allows the equation to be written as \((x-5)(x+4) = 0\).

Step 5: Solve for x

Now, you can solve for x. Setting each factor equal to zero gives the solutions \(x = 5\) and \(x = -4\).

Step 6: Checking the solutions

Finally, you need to check the solutions into the original equation. Plugging in \(x = 5\) into the original equation gives \(6 = 6\) which is true. However, plugging in \(x = -4\) into the original equation doesn't satisfy the original equation. So, \(x = -4\) is excluded.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental building block in algebra. They are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are constants and \( a \) is not equal to zero. In the given exercise, the equation \( x^2 - x - 20 = 0 \) is a classic example of a quadratic equation.

Solving a quadratic equation involves finding the values of \( x \) that make the equation true. There are multiple methods to achieve this, including factoring, completing the square, or using the quadratic formula. The method chosen depends on the particular form and complexity of the equation.

In this exercise, upon isolating the expression and simplifying, we reach a form suitable for factoring, which is the most straightforward method when applicable. Recognizing and solving quadratic equations is essential when dealing with more complex problems involving algebraic expressions.

Factoring Polynomials

Factoring polynomials is a key technique in algebra that simplifies complex expressions and is often used to solve equations. This process involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give back the original polynomial.

For the quadratic equation at hand, factoring meant representing \( x^2 - x - 20 \) as \( (x - 5)(x + 4) \) which is the product of two binomials. To successfully factor a quadratic, you need to find two numbers that multiply to give the constant term \( c \), and add up to the linear coefficient \( b \).

Steps to Factoring a Quadratic Polynomial:

• Identify \( a \) (coefficient of \( x^2 \) term), \( b \) (coefficient of \( x \) term), and \( c \) (constant term).
• Find two numbers that multiply to \( ac \) and sum to \( b \) (if \( a = 1 \) it's just the constant term \( c \) and the linear coefficient \( b \)).
• Rewrite the quadratic polynomial using these two numbers to split the middle term.
• Factor by grouping, if needed, to find the binomial factors.

Once a polynomial is factored, solving the corresponding equation often becomes straightforward, as seen in this instructional exercise.

Exponent Rules

Understanding exponent rules is vital when working with algebraic expressions, especially when manipulating equations with rational exponents. Rational exponents represent roots and powers combined and are written as a fraction \( a^{m/n} \) where \( a \) is the base, \( m \) is the exponent, and \( n \) is the index of the root.

Key exponent rules applied in this exercise include:

• The Power rule: \( (a^{m})^{n} = a^{mn} \) states that when raising a power to another power, you multiply the exponents.
• The Root rule: \( a^{1/n} = \sqrt[n]{a} \) converts a rational exponent to a radical form.
• Raising both sides of an equation to the same power does not change the equality (if both sides are non-negative).

By raising each side of our equation \( (x^2 - x - 4)^{3/4} = 8 \) to the power of \( 4/3 \) we eliminated the rational exponent, transforming the equation into a simpler quadratic form. Understanding how to handle exponents—whether integer or rational—is crucial for algebra students as it frequently comes into play in various types of mathematical problems.