Problem 65 - Algebra for College Students

Question

Solve each of the equations. $$x^{2}+7 x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -7\).

Step 1: Identify the structure of the equation

The equation is quadratic and can be written as \(x^2 + 7x = 0\). It is in the form of \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 7\), and \(c = 0\).

Step 2: Factor the equation

Notice that \(x\) is a common factor. Factor the equation as \(x(x + 7) = 0\).

Step 3: Apply the Zero Product Property

According to the Zero Product Property, if a product of two numbers is zero, then at least one of the numbers must be zero. Set each factor of the equation equal to zero: \(x = 0\) and \(x + 7 = 0\).

Step 4: Solve the individual equations

The first equation is \(x = 0\). The second equation is \(x + 7 = 0\). Solve for \(x\) by subtracting 7 from both sides, giving you \(x = -7\).

Step 5: Conclude the solution

The solutions to the equation \(x^2 + 7x = 0\) are \(x = 0\) and \(x = -7\).

Key Concepts

Factoring

Factoring is a friendly technique used to break down equations into simpler parts. It's particularly handy for quadratic equations like our example \(x^2 + 7x = 0\). Here's how it works:

When you look at the equation, you'll notice that both terms, \(x^2\) and \(7x\), share a common factor, which is \(x\). This means you can "take out" \(x\) from both terms, simplifying the expression. By factoring, the equation becomes \(x(x + 7) = 0\).

Factoring allows us to express the equation as a product of its factors. This step is essential because it sets the stage for using the Zero Product Property, which we'll dive into next.

Remember, the first step in factoring is always to look for a common factor in all parts of the equation. Not only does this simplify the equation, but it also makes finding solutions much easier.

Zero Product Property

The Zero Product Property is a simple but powerful concept. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is used in mathematics to solve factored equations like \(x(x + 7) = 0\).

Moving forward, you use this property to solve the factored form of the quadratic equation. Since you have \(x\) times \((x + 7)\) equal to zero, either \(x = 0\) or \(x + 7 = 0\) must be true.

Busting it down:

• If \(x = 0\), then our equation is true.
• If \(x + 7 = 0\), we can solve for \(x\) to find another solution.

By applying this property, you can determine the values of \(x\) that satisfy the original equation. This is a critical step in solving quadratic equations through factoring.

Solving Equations

Once an equation is factored and you're utilizing the Zero Product Property, solving becomes the fun part. In our example, after factoring \(x(x + 7) = 0\), you've realized either \(x = 0\) or \(x + 7 = 0\).

Let's look at these individually:

• For \(x = 0\), there's no further math needed. \(x = 0\) is your first solution.
• For \(x + 7 = 0\), subtract 7 from both sides to isolate \(x\), yielding \(x = -7\).

You've now effectively solved the quadratic equation, finding that \(x\) can be either 0 or -7. Thus, the solutions to the equation \(x^2 + 7x = 0\) are \(x = 0\) and \(x = -7\).

Remember, breaking the solution process into smaller parts: factoring, using the Zero Product Property, and solving each part, makes even the toughest equations much easier to handle.