Problem 3 - Beginning and Intermediate Algebra

Question

Solve each equation.. \((m+9)(m-8)=0\)

Step-by-Step Solution

Answer

The values of \(m\) that satisfy the equation \((m+9)(m-8) = 0\) are \(m=-9\) and \(m=8\).

Step 1: Apply the zero-product property

Since the product of the two factors \((m+9)(m-8)\) is equal to zero, we can set each factor equal to zero. So, we have: \(m+9=0\) and \(m-8=0\)

Step 2: Solve the first equation

Solving the equation \(m + 9 = 0\) for \(m\): Subtract 9 from both sides of the equation: \(m=-9\)

Step 3: Solve the second equation

Solving the equation \(m - 8 = 0\) for \(m\): Add 8 to both sides of the equation: \(m=8\)

Step 4: Combine the solutions

We found the solutions \(m=-9\) and \(m=8\). These are the two values of \(m\) that will satisfy the quadratic equation \((m+9)(m-8) = 0\).

Key Concepts

Zero-Product Property

The zero-product property is a fundamental concept in algebra that helps us solve quadratic equations. When two numbers multiply to zero, at least one of them must be zero. This means if you have an equation like

• \[(a)(b) = 0\]

Either \(a = 0\) or \(b = 0\) (or both). This property is powerful because it allows us to break down more complex equations into simpler ones.
For instance, in our example,

• \((m+9)(m-8)=0\):

We can set each factor to zero individually and solve the resulting linear equations. This leads directly to the solutions of the equation.

Factoring

Factoring is the process of breaking down an expression into products of simpler expressions, which we call factors. This is a key step in solving quadratic equations. By factoring a quadratic equation, we can use the zero-product property to find the variable values that make the equation true.
For example, given a quadratic in the form of

• \[ax^2 + bx + c = 0\]

We aim to rewrite it as

• \[(x + p)(x + q) = 0\]

Where \(p\) and \(q\) are numbers that make the equation easier to solve. In the exercise,

• \((m+9)(m-8)\)

is already factored, simplifying the process of solving the equation.

Solving Equations

Solving equations involves finding the value(s) of the unknown variable(s) that make the equation true. In our context, we are solving for \(m\) in the factored quadratic equation

• \((m+9)(m-8) = 0\)

Once we apply the zero-product property and set each factor to zero, we solve each resultant equation.
For

• \(m + 9 = 0\)

We subtract 9 from both sides, yielding

• \(m = -9\)

Similarly, for

• \(m - 8 = 0\)

We add 8 to both sides, finding

• \(m = 8\)

These calculations show that solving equations is often about applying simple operations to find variable values.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It involves solving equations and understanding how different equations represent relationships between quantities. Quadratic equations, like the one in our exercise, are a common type of algebraic equation, characterized by containing an unknown raised to the second power.

• The general form is \[ax^2 + bx + c = 0\]

Quadratics can often be solved using methods such as factoring, completing the square, or using the quadratic formula. Our exercise centered on using factoring to solve a quadratic equation, showcasing one practical application of algebra.