Problem 353 - Elementary Algebra

Question

Solve. \(m^{2}=6 m+16\)

Step-by-Step Solution

Answer

The solutions for are .

Step 1 - Move All Terms to One Side

Rewrite the given equation so that all terms are on one side of the equation, forming a standard quadratic equation:

Step 2 - Set the Equation to Zero

Subtract both from both sides: Now the equation is Finally combine like terms to end up with:

Step 3 - Factor the Quadratic Equation

factor the equation to: , .

Step 4 - Solve for 'm'

Set each factor equal to zero and solve for value of 'm' i.e., solve and

Step 5 - Write Final Answer

Now write the final solutions:

Key Concepts

quadratic equations

A quadratic equation is an equation that takes the form: \(ax^2 + bx + c = 0\). In the given exercise, our equation starts as: \(m^2 = 6m + 16\).To recognize it as a quadratic equation, we need to rearrange it into the standard quadratic form.This involves moving all terms to one side of the equation.First, subtract \(6m\) and \(16\) from both sides:\(m^2 - 6m - 16 = 0\). Now, we have a standard quadratic equation.

factoring equations

Factoring a quadratic equation is useful as it breaks down the equation into simpler parts. We look for two numbers that multiply to give us the constant term \(-16\) and add to give the middle coefficient \(-6\). Think of factors like this:

• Product (multiplication) of numbers = \(-16\)
• Sum (addition) of numbers = \(-6\)

Here, the numbers that meet these requirements are \(-8\) and \(2\). Thus, the equation can be factored as:\((m - 8)(m + 2) = 0\) .

algebraic solutions

Once we have factored the quadratic equation, solving for the variable involves setting each factor to zero. For the equation \((m - 8)(m + 2) = 0\), we set each factor to zero:\(m - 8 = 0\)and \(m + 2 = 0\). Solving these equations:

• \(m = 8\)
• \(m = -2\)

These solutions (values) are the roots of the quadratic equation.

elementary algebra

Elementary algebra involves basic techniques for solving equations. In this context, we're dealing with moving terms across an equation and factoring. Here’s a recap:

• Move all terms to one side to form a standard quadratic equation.
• Factor the equation into simpler terms.
• Solve each simpler equation derived from factors.

Our original problem \(m^2 = 6m + 16\) transforms to \(m - 8 = 0\) and \(m + 2 = 0\), and the solutions are \(m = 8\) and \(m = -2\). This process showcases the blend of fundamental algebraic skills.