Problem 15 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ (x+6)(x-1)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are x = {-6, 1}.

Step 1: Apply the zero-product property

We have the factored equation (x+6)(x-1)=0. By the zero-product property, either (x+6)=0 or (x-1)=0.

Step 2: Solve the first linear equation

First, we solve the equation (x+6)=0. Subtract 6 from both sides to isolate x: x=-6.

Step 3: Solve the second linear equation

Now, we solve the equation (x-1)=0. Add 1 to both sides to isolate x: x=1.

Step 4: Write the final solution

Since we found x=-6 and x=1, the complete solution to the given quadratic equation is x = {-6, 1}.

Key Concepts

Zero-Product Property

The zero-product property is a crucial concept in solving equations involving the multiplication of terms. The property states that if the product of two numbers is zero, then at least one of the numbers must be zero. This seems simple, but it's very powerful in solving quadratic equations that are factored into two linear terms.

For example, consider the equation \((x+6)(x-1)=0\). Here, you have two terms being multiplied. According to the zero-product property, for the product \((x+6)(x-1)\) to be zero, it means:

• Either \(x+6=0\)
• Or \(x-1=0\)

Now, each of these is a simple linear equation that can be solved individually, making the entire process much simpler.

Factoring

Factoring is a mathematical process used to express an equation as a product of its terms. This is especially helpful when you are dealing with quadratic equations. A quadratic equation is normally given in the form \(ax^2 + bx + c = 0\).

The goal in factoring is to rewrite this equation in a form such as \((x+p)(x+q)\). Once the quadratic is factored, you can apply the zero-product property as shown above.

• Start by identifying factors of the constant and middle terms that add up to the middle term's coefficient.
• Use these factors to write the equation in a factored form.

In our example, the equation \((x+6)(x-1)=0\) is already factored, so you can proceed directly to applying the zero-product property. But sometimes, you might need to factor a quadratic expression before using it.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. Once you have used factoring and the zero-product property to break down a quadratic equation, what's left are linear equations to solve.

Let's look at the equation \((x+6)=0\):

• Isolate the variable by performing the same operation on both sides of the equation.
• Subtract 6 from both sides to find \(x = -6\).

Next, consider the equation \((x-1)=0\):

• Again, isolate the variable by performing the same operation on both sides.
• Add 1 to both sides to get \(x = 1\).

Once both linear equations are solved, you have discovered the solutions to the original quadratic equation, which are \(x = -6\) and \(x = 1\). Understanding how to solve these simple linear equations is key in finding solutions to more complex problems.