Problem 39 - Intermediate Algebra

Question

Solve each equation. See Example 5. $$ \left(m^{3}+26\right)^{1 / 3}=m+2 $$

Step-by-Step Solution

Answer

The solutions are \( m = -3 \) and \( m = 1 \).

Step 1: Understand the Given Equation

The equation provided is \( \left(m^{3}+26\right)^{1 / 3}=m+2 \). Here, you need to solve for the variable \( m \). The left side of the equation involves a cube root, and the goal is to isolate \( m \) to find its value.

Step 2: Eliminate the Cube Root

To eliminate the cube root on the left side of the equation, cube both sides. This gives \[(m^{3} + 26)^{1/3})^{3} = (m + 2)^{3}.\]Simplifying the left side, we have:\[m^{3} + 26 = (m+2)^{3}.\]

Step 3: Expand the Cubic Term on the Right

Expand the right side of the equation:\[(m+2)^{3} = m^{3} + 3m^{2} \times 2 + 3m \times 2^{2} + 2^{3} = m^{3} + 6m^{2} + 12m + 8.\]So, the equation becomes:\[m^{3} + 26 = m^{3} + 6m^{2} + 12m + 8.\]

Step 4: Simplify the Equation

Subtract \( m^{3} \) from both sides:\[26 = 6m^{2} + 12m + 8.\]

Step 5: Rearrange into Standard Quadratic Form

Subtract 8 from both sides to get:\[26 - 8 = 6m^{2} + 12m\]which simplifies to\[18 = 6m^{2} + 12m.\]

Step 6: Solve the Quadratic Equation

To solve \( 6m^{2} + 12m - 18 = 0 \), first divide the entire equation by 6 to simplify:\[m^{2} + 2m - 3 = 0.\]Factor the quadratic equation:\[(m+3)(m-1) = 0.\]

Step 7: Find the Solutions for m

Set each factor to zero and solve for \( m \):- \( m + 3 = 0 \) gives \( m = -3 \).- \( m - 1 = 0 \) gives \( m = 1 \).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear frequently in different math areas. A quadratic equation takes the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable to be solved. The equation is called "quadratic" because the highest degree term is squared, i.e., \( x^2 \).

Solving quadratic equations is essential for higher-level math concepts. Generally, there are three main methods to solve them: factoring, completing the square, and using the quadratic formula.

• **Factoring**: This technique involves expressing the quadratic equation as a product of two binomials. For example, \( (m+3)(m-1) = 0 \) is the factored form of the equation \( m^2 + 2m - 3 = 0 \).


• **Completing the Square**: This involves rearranging the equation to form a perfect square trinomial, which can then be solved easily.


• **Quadratic Formula**: This is a surefire method to find solutions for any quadratic equation using the formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Quadratic equations often have two solutions since they are based on a squared term. Understanding these solutions requires a grasp of algebraic manipulation and recognizing patterns in equations.

Factoring

Factoring is a powerful algebraic technique used to simplify expressions and solve equations. Essentially, it involves breaking down a complex expression into simpler, multiplied factors. When we talk about solving quadratic equations, factoring is often the first method explored because of its simplicity when applicable.

Consider an equation in the form \( m^2 + 2m - 3 = 0 \). To factor this, you look for two numbers that multiply to \( -3 \) (the constant term) and add up to \( 2 \) (the linear coefficient). These numbers are \( -1 \) and \( 3 \), leading to the factorization \( (m+3)(m-1) = 0 \).

• **Check your work**: Always expand the factors to ensure they multiply back to the original quadratic expression. This ensures your factoring is correct.


• **Zero Product Property**: Once factored, use this property to set each factor equal to zero: \( m+3=0 \) and \( m-1=0 \). This allows for finding the values of \( m \) that solve the equation.

Factoring is greatly beneficial when quick and simple solutions are needed, and it provides a deeper understanding of the equation's structure.

Cube Roots

A cube root is the number that, when cubed (multiplied by itself twice), results in the original number under the cube root. Mathematically, the cube root of \( a \) is represented as \( a^{1/3} \). In this exercise, you see a cube root in the equation \( (m^3 + 26)^{1/3} = m + 2 \).

To solve an equation involving a cube root, the typical approach is to eliminate the cube root by cubing both sides of the equation, as seen in the step from the original solution:

• This transforms \( (m^3 + 26)^{1/3} \) into \( m^3 + 26 \).


• Likewise, the right side \( (m + 2) \) becomes \( (m + 2)^3 \), which simplifies to \( m^3 + 6m^2 + 12m + 8 \) after expansion.

Cubing both sides helps simplify the problem by removing the cube root, allowing for further algebraic manipulation. Understanding cube roots and how to manipulate them is crucial in solving such equations, making this concept an important tool in your math toolkit.