Problem 37 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt[3]{x^{3}-7}=x-1 $$

Step-by-Step Solution

Answer

The equation \( \sqrt[3]{x^3 - 7} = x - 1 \) has no real solutions because the discriminant is negative.

Step 1: Isolate the radical expression

Start by isolating the cube root term on one side of the equation. The given equation is \( \sqrt[3]{x^{3} - 7} = x - 1 \). The cube root is already isolated in this case.

Step 2: Eliminate the cube root

To remove the cube root, cube both sides of the equation. This gives: \( x^3 - 7 = (x - 1)^3 \).

Step 3: Expand the right side

Expand the cubic expression on the right side of the equation. \( (x - 1)^3 \) expands to \( x^3 - 3x^2 + 3x - 1 \).

Step 4: Simplify the equation

Substitute the expanded form back into the equation: \( x^3 - 7 = x^3 - 3x^2 + 3x - 1 \).

Step 5: Solve for zero

Subtract \( x^3 \) from both sides: \( -7 = -3x^2 + 3x - 1 \). Add terms to set the equation to zero: \( 0 = 3x^2 - 3x + 6 \).

Step 6: Divide through by a common factor

To simplify the equation, divide every term by 3: \( 0 = x^2 - x + 2 \).

Step 7: Use the quadratic formula

Use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = 2 \). Substitute these values into the formula: \( x = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(2)}}{2(1)} \).

Step 8: Calculate the discriminant

Compute the discriminant \( b^2 - 4ac \): \( 1 - 8 = -7 \). Since the discriminant is negative, \( -7 \), the equation has no real solutions.

Key Concepts

Cubic Equations

Cubic equations are algebraic equations where the highest degree is three. This means they are of the form \( ax^3 + bx^2 + cx + d = 0 \) where \( a eq 0 \). In our exercise, the equation involves a cubic root, which simplifies the expression but initially looks complex. To solve such an equation, it's crucial to eliminate the cube root first by cubing both sides. This process transforms the equation into something more familiar, making it easier to apply further algebraic techniques, such as expansion of products.

Quadratic Formula

The quadratic formula is a fundamental tool for finding the roots of a quadratic equation. Quadratic equations take the form \( ax^2 + bx + c = 0 \). Here, the formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Using this formula requires identifying the coefficients \( a \), \( b \), and \( c \) from the equation. It's a handy method when an equation doesn't easily factorize, allowing us to determine the roots through direct computation.

Discriminant

The discriminant is an essential component of the quadratic formula. It is the expression \( b^2 - 4ac \) under the square root in the formula. The value of the discriminant helps us determine the nature of the roots:

• If it is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, or the roots are equal.
• If it is negative, as in our exercise with a discriminant of \(-7\), the equation has no real solutions, implying complex roots.

Understanding the discriminant helps in quickly predicting the type of solutions without solving entirely.

Radical Expressions

In algebra, radical expressions involve roots, such as square roots or cube roots. The technique of isolating a radical expression involves arranging your equation so the radical stands alone on one side. This was the first step in solving our equation. Following isolation, raising both sides of the equation to the power that matches the root value clears the radical, which simplifies solving. It's important to handle radicals carefully because they can significantly change the equation when improperly managed.