Problem 20 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 5 x(5 x+7)=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = -\frac{7}{5} \).

Step 1: Understand the Equation

The equation given is \( 5x(5x + 7) = 0 \). It is a factorization of the form \( a \times b = 0 \), where \( a = 5x \) and \( b = 5x + 7 \). To solve it, we apply the Zero Product Property.

Step 2: Apply the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, at least one of the factors must be zero. Thus, set each factor equal to zero: \( 5x = 0 \) and \( 5x + 7 = 0 \).

Step 3: Solve the First Equation

Solve the first factor equation \( 5x = 0 \). Divide both sides by 5 to isolate \( x \):\[ x = \frac{0}{5} = 0 \].

Step 4: Solve the Second Equation

Now, solve the second factor equation \( 5x + 7 = 0 \). Subtract 7 from both sides:\[ 5x = -7 \].Then divide both sides by 5 to solve for \( x \):\[ x = -\frac{7}{5} \].

Step 5: State the Solutions

The solutions to the equation \( 5x(5x + 7) = 0 \) are \( x = 0 \) and \( x = -\frac{7}{5} \).

Key Concepts

Zero Product Property

When you face an equation that involves a product equaling zero, the Zero Product Property is your friend. This property tells us that if a product of two numbers is zero, then one or both numbers must be zero. For instance, if \(a \times b = 0\), then either \(a = 0\) or \(b = 0\), or both.
This principle is vital when solving quadratic equations because it allows us to break down complex equations into simpler parts. By isolating each factor and setting them individually to zero, we can find possible solutions for the variable.
Using this property effectively can simplify what initially seems like a complicated problem. It neatly splits the task of solving into manageable steps, making it a fundamental tool in algebra.

Factoring

Factoring is a classic technique used to solve quadratic equations. It involves expressing an equation as a product of its simpler components, called factors.
In the case of the equation \(5x(5x + 7) = 0\), this step is already given as it is already factorized as \((5x)\) and \((5x + 7)\).
Being able to factor equations allows you to use the Zero Product Property efficiently. If an equation is in a format where the product equals zero, you can directly apply the property. But first, you need each equation part's factors.
Remember, not all equations are ready to be factored at first glance. Some can be rearranged or simplified, turning them into a more favorable form for factoring, thus making solutions easier to find.

Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. Once we've factored an equation, like in this exercise, it becomes much simpler.
Here’s how you solve our example step by step:

• Identify the factors: You have \(5x\) and \(5x + 7\).
• Set each factor to zero: \(5x = 0\) and \(5x + 7 = 0\).

For \(5x = 0\):

• Divide both sides by 5 to get \(x = 0\).

For \(5x + 7 = 0\):

• Subtract 7 from both sides to get \(5x = -7\), then divide by 5 to solve for \(x\), resulting in \(x = -\frac{7}{5}\).

You now have your solutions: \(x = 0\) and \(x = -\frac{7}{5}\). These steps underline how effectively applying known properties and techniques can lead to the right answers.