Problem 25 - Elementary Algebra

Question

For the following problems, solve the equations. $$ m^{2}-81=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(m^2 - 81 = 0\) for the values of m. Answer: The solutions to the given quadratic equation are \(m = -9\) and \(m = 9\).

Step 1: Identify the given quadratic equation

We're given the quadratic equation: $$ m^{2} - 81 = 0 $$

Step 2: Factor using difference of squares formula

Now, express the given equation in the form of difference of squares since 81 is a perfect square: $$ m^2 - 81 = (m + 9)(m - 9) $$

Step 3: Solve for m

Now, to solve the equation, equate each factor to zero and solve for m: 1) \((m + 9) = 0\): $$ m = -9 $$ 2) \((m - 9) = 0\): $$ m = 9 $$

Step 4: Present the solution

The solutions to the given quadratic equation are: $$ m = -9, \, 9 $$

Key Concepts

Factoring Polynomials

Factoring polynomials is a method used to express a polynomial as the product of its simplest parts, known as factors. For quadratic equations, factored forms are easier to solve, particularly when finding the roots of the equation. In this exercise, the quadratic polynomial is given by \( m^2 - 81 \). The goal is to rewrite this equation as a multiplication statement, using its factors.

• Identify parts of the polynomial that can be rewritten in simpler forms.
• Factors are typically binomials, like \((m + a)(m - a)\) that when multiplied yield the original polynomial.
• This technique breaks down the polynomial into manageable equations.

Factoring is a cornerstone skill in algebra because it is simple, effective, and allows for solving complex expressions using easier calculations. Once we find the factors, solving the equation means setting each factor equal to zero and finding the unknown variable.

Difference of Squares

The difference of squares is a specific algebraic identity used in factoring. It comes into play when you encounter a quadratic expression in the form \( a^2 - b^2 \). This identity states that \( a^2 - b^2 = (a + b)(a - b) \). Understanding this concept is crucial when dealing with quadratic equations that fit this form.

• The expression \( m^2 - 81 \) is recognized as a difference of squares.
• Here, \( a^2 = m^2 \) and \( b^2 = 81 \), hence \( a = m \) and \( b = 9 \).
• Applying the difference of squares identity, \( m^2 - 81 \) factors into \( (m + 9)(m - 9) \).

Factoring using the difference of squares allows us to simplify and solve the quadratic equation easily. This is because the equation is expressed in a form that highlights its roots immediately when setting each factor to zero.

Algebraic Solutions

Algebraic solutions refer to the process of solving equations using algebraic operations and methods. In this scenario, after factoring the polynomial using the difference of squares, we proceed to find the values of \( m \) that solve \( m^2 - 81 = 0 \).

• Write down the factored equation: \((m + 9)(m - 9) = 0\).
• Apply the zero-product property: for \((x)(y) = 0\), either \(x = 0\) or \(y = 0\).
• Set each factor to zero: \(m + 9 = 0\) or \(m - 9 = 0\).
• Solve each simple equation: \(m = -9\) or \(m = 9\).

These solutions, \(m = -9\) and \(m = 9\), are the roots of the original quadratic equation. Mastering algebraic solutions involves understanding both how to manipulate equations and how to implement factoring and the properties of numbers efficiently. This skill aids in tackling a wide array of algebraic problems.