Problem 38 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ m^{2}-49=0 $$

Step-by-Step Solution

Answer

The solutions are \( m = 7 \) and \( m = -7 \).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \( m^2 - 49 = 0 \). This is a difference of squares.

Step 2: Apply the Formula for Difference of Squares

Recall the difference of squares formula: \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a = m \) and \( b = 7 \), since \( 49 \) is \( 7^2 \). Therefore, we can express the equation as \((m-7)(m+7) = 0 \).

Step 3: Solve Each Factor Separately

Set each factor equal to zero and solve for \( m \):1. \( m - 7 = 0 \Rightarrow m = 7 \)2. \( m + 7 = 0 \Rightarrow m = -7 \)

Step 4: Combine the Solutions

The solutions to the equation \( m^2 - 49 = 0 \) are \( m = 7 \) and \( m = -7 \).

Key Concepts

Difference of Squares

In algebra, the difference of squares is a useful technique to solve equations of the type \( a^2 - b^2 = 0 \). This technique is particularly handy because it lets us break down complex equations into simpler binomial expressions. Consider the equation from our exercise, \( m^2 - 49 = 0 \). Here, the goal is to recognize it as a difference of squares. To identify it, note that 49 can be rewritten as \( 7^2 \). We now have the expression \( m^2 - 7^2 \), complying with the pattern of difference of squares \( a^2 - b^2 = (a-b)(a+b) \).

In our exercise, \( a \) is \( m \) and \( b \) is 7. Substituting these back into the formula gives us the expression \((m-7)(m+7) = 0\). Breaking equations into factors simplifies the solving process, which leads us to the next concept: factoring.

Factoring

Factoring is the process of rewriting an expression as a product of its factors. This is beneficial in solving quadratic equations as it converts them into easier, solvable parts. In our quadratic equation \( m^2 - 49 = 0 \), after identifying it as a difference of squares, we factor it into two binomials: \((m-7)(m+7)=0\).

Think of factoring this way: you are breaking down the equation into 'building blocks' that multiply to form the original quadratic expression.

• First, identify the quadratic equation and verify if it fits a specific factorable pattern.
• Next, apply the appropriate formula to express it in factored form.
• Finally, solve these simpler equations.

Factoring these expressions is useful as it helps in revealing the solutions to the equation quickly and efficiently. It prepares us for the final step, which is solving each of the factored terms.

Solving Equations

Once we have factored the quadratic equation, solving involves finding the variable's possible values. These are known as the equation's roots. For our example, we transformed \( m^2 - 49 = 0 \) into \((m-7)(m+7)=0\). From here, set each factor equal to zero:

\( m - 7 = 0 \) and \( m + 7 = 0 \).

This gives us two simple equations to solve:

• For \( m - 7 = 0 \), add 7 to both sides to get \( m = 7 \).
• For \( m + 7 = 0 \), subtract 7 to isolate \( m \), yielding \( m = -7 \).

Solving these linear equations tells you that the solutions to the original quadratic are \( m = 7 \) and \( m = -7 \). Thus, factoring and the properties of the difference of squares simplify solving quadratic equations considerably.