Problem 74 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 2 m^{2}-8 m=2 m-12 $$

Step-by-Step Solution

Answer

The solutions are \( m = 2 \) and \( m = 3 \).

Step 1: Move Terms to One Side

Start solving the equation by moving all terms to one side of the equation to set it to zero. We want to subtract \(2m\) from both sides.\[2m^2 - 8m - 2m = 2m - 2m - 12\]This simplifies to:\[2m^2 - 10m + 12 = 0\]

Step 2: Simplify the Equation

Check if the equation can be simplified by dividing through by a common factor. Here, each term can be divided by 2:\[m^2 - 5m + 6 = 0\]

Step 3: Factor the Quadratic Equation

Now, factor the quadratic equation \(m^2 - 5m + 6 = 0\). Look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3, so the equation becomes:\[(m - 2)(m - 3) = 0\]

Step 4: Solve for \(m\)

To find the solutions, set each factor equal to zero:- \(m - 2 = 0\) which gives \(m = 2\).- \(m - 3 = 0\) which gives \(m = 3\).Thus, the solutions are \(m = 2\) and \(m = 3\).

Key Concepts

Factoring

Factoring is an essential concept in solving quadratic equations. It involves expressing an equation as a product of simpler expressions. When we factor a quadratic equation, we are essentially undoing the multiplication of two binomials. For example, given the quadratic equation \(m^2 - 5m + 6 = 0\), we aim to find two binomials whose product equals this expression.

The process begins by identifying two numbers that multiply to the constant term, which is 6 in this case, and add up to the coefficient of the linear term, which is -5. In this example, these numbers are -2 and -3.

• The equation \(m^2 - 5m + 6\) factors into \((m - 2)(m - 3)\).
• Each factor represents a solution when set to zero.

Factoring requires practice to recognize patterns quickly. It's often used when solving quadratic equations as it simplifies finding solutions.

Solving Equations

Solving quadratic equations involves finding the values of the variable that make the equation true. Once we have factored the quadratic equation, the next step is to solve it. In our example, after factoring \((m - 2)(m - 3) = 0\), we use the principle that if a product of numbers is zero, then at least one of the numbers must be zero.

By setting each factor to zero, we solve the simple linear equations:

• \(m - 2 = 0\) gives \(m = 2\).
• \(m - 3 = 0\) gives \(m = 3\).

These solutions are the values of \(m\) that satisfy the original quadratic equation. Solving these smaller linear equations is straightforward, mostly involving basic arithmetic.

Algebraic Expressions

An algebraic expression is composed of numbers, variables, and arithmetic operations. Understanding algebraic expressions is key in manipulating and solving equations.

In solving quadratic equations, we often work with expressions that can be simplified or reorganized. Consider the original expression \(2m^2 - 8m = 2m - 12\). It needs to be arranged so that all terms are on one side, leading to a standard form for quadratic equations, which is \(ax^2 + bx + c = 0\).

• First, move all terms to one side: \(2m^2 - 10m + 12 = 0\).
• Then, simplify by dividing each term by 2: \(m^2 - 5m + 6 = 0\).

By rewriting the equation in its simplest form, it's easier to apply methods like factoring to solve it. A good grasp of algebraic expression manipulation is crucial for effective problem-solving in mathematics.