Problem 12 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$m^{2}+2 m=8$$

Step-by-Step Solution

Answer

The solutions are \(m = 2\) and \(m = -4\).

Step 1: Rewrite the Equation in Standard Form

First, rewrite the equation in standard form by setting it equal to zero. The given equation is: \[ m^{2} + 2m = 8 \]. Subtract 8 from both sides to get: \[ m^{2} + 2m - 8 = 0 \]

Step 2: Identify the Coefficients

Identify the coefficients of the quadratic equation in the form \(ax^2 + bx + c = 0\). Here, we have: \[ a = 1, b = 2, c = -8 \]

Step 3: Write Down the Quadratic Formula

Recall the quadratic formula, which is: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 4: Substitute the Coefficients into the Formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \[ m = \frac{-(2) \pm \sqrt{(2)^2 - 4(1)(-8)}}{2(1)} \] Simplify inside the square root: \[ m = \frac{-2 \pm \sqrt{4 + 32}}{2} \] \[ m = \frac{-2 \pm \sqrt{36}}{2} \]

Step 5: Simplify the Square Root

Simplify \(\sqrt{36}\): \[ m = \frac{-2 \pm 6}{2} \]

Step 6: Solve for m

Now, solve for \(m\) by splitting into two solutions: \[ m = \frac{-2 + 6}{2} = 2 \] and \[ m = \frac{-2 - 6}{2} = -4 \]

Key Concepts

solving quadratic equations

Solving quadratic equations can seem daunting, but understanding the core principles simplifies the process. A quadratic equation typically looks like this: \[ ax^2 + bx + c = 0 \]where

• a (leading coefficient) is the coefficient of the term with the variable squared
• b is the coefficient of the term with just the variable
• c is the constant term (without any variable)

In the standard form, the equation is always set to zero. This positioning enables us to use formulas or other methods to find the values of the variable that satisfy the equation. The provided exercise first places the equation in standard form before applying the quadratic formula. Rewriting the initial equation and simplifying the terms are essential first steps.

quadratic formula

The quadratic formula is a powerful tool for solving any quadratic equation. The formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] provides solutions for the variable x. To use this formula, you simply plug in the values of the coefficients a, b, and c into their respective positions.

Let's break it down:

• -b: The formula begins with the negation of the coefficient b.
• \( \pm \): This symbol means that there will be two solutions; one involves addition and the other involves subtraction.
• \( \sqrt{b^2 - 4ac} \): This part of the formula calculates the discriminant. It determines the nature of the roots (real or complex).
• \(2a\): Finally, you divide the entire expression by twice the coefficient of the squared term.

In the provided solution, the coefficients were substituted into this formula, and the quadratic formula was used step-by-step to solve for the variable m.

algebra

Algebra is the broad area of mathematics dealing with symbols and the rules for manipulating those symbols. In solving quadratic equations like \[ m^2 + 2m = 8 \], we use algebraic techniques to rearrange and simplify the equation.

Here are some key algebraic steps involved:

• Rewriting: Bringing the equation to its standard form by setting it to zero, for instance, by subtracting 8 from both sides in our given exercise.
• Identifying Coefficients: Recognizing the values of a, b, and c, which are crucial for the quadratic formula.
• Simplifying Expressions: Performing arithmetic to simplify expressions within the formula, such as finding square roots or performing division.
• Finding Solutions: Splitting the equation into two parts (one using + and the other using -), solving for the variable in both cases.

Through these algebraic methods, you can solve various types of equations, making algebra an indispensable part of mathematics.