Problem 50 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 3 s^{2}=-4 s $$

Step-by-Step Solution

Answer

The solutions are \(s = 0\) and \(s = -\frac{4}{3}\).

Step 1: Set Equation to Zero

First, move all terms to one side of the equation to set it to zero: \[ 3s^2 + 4s = 0 \]

Step 2: Factor the Equation

Factor out the common term \(s\):\[ s(3s + 4) = 0 \]

Step 3: Apply the Zero Product Property

Using the zero product property, set each factor equal to zero:1. \( s = 0 \)2. \( 3s + 4 = 0 \)

Step 4: Solve Each Equation

Solve for \(s\) from each factor:1. From \( s = 0 \), we have \( s = 0 \).2. From \( 3s + 4 = 0 \), solve for \(s\) by rearranging and simplifying: \[ 3s = -4 \ s = -\frac{4}{3} \]

Step 5: State the Solutions

The solutions to the equation \(3s^2 = -4s\) are:\(s = 0\) and \(s = -\frac{4}{3}\).

Key Concepts

Setting Equation to Zero

In order to solve quadratic equations, it's important to first set the equation to zero. This means that all terms of the equation are on one side, with the equation equating to zero on the other side. This step is critical because it prepares the equation for factoring or other methods of solving.

For example, given the equation \(3s^2 = -4s\), you need to move the \(-4s\) term to the left side of the equation. You do this by adding \(4s\) to both sides:

• Original equation: \(3s^2 = -4s\)
• Move \(-4s\) to the left: \(3s^2 + 4s = 0\)

By doing this, you have successfully set the equation to zero and you are now ready to apply further techniques like factoring. This step ensures that the equation is in a solvable format.

Factoring

Factoring is a helpful technique used to simplify and solve quadratic equations. It involves expressing the equation as a product of its factors, which are simpler expressions that multiply to create the original equation.

In the example \(3s^2 + 4s = 0\), you can factor out the common factor, which is \(s\), from each term:

• Identify common factor: Both terms \(3s^2\) and \(4s\) have the factor \(s\).
• Factor the equation: \(s(3s + 4) = 0\)

By factoring, you break the equation into components that can each independently solve the equation. This makes it significantly easier to find solutions, as you can then use each factor to apply the Zero Product Property.

Zero Product Property

The Zero Product Property is a key principle used in solving quadratic equations. It states that if a product of two numbers is zero, then at least one of the numbers must be zero. After setting the equation to zero and factoring, this property is used to find solutions.

When you have factored the equation \(3s^2 + 4s = 0\) into \(s(3s + 4) = 0\), the Zero Product Property allows you to set each factor to zero:

• \(s = 0\)
• \(3s + 4 = 0\)

By solving each factor separately, you find that:

• For \(s = 0\), the solution is simply \(s = 0\).
• For \(3s + 4 = 0\), solve for \(s\) by subtracting 4 from both sides and dividing by 3: \(s = -\frac{4}{3}\).

This dual solution method enabled by the Zero Product Property is crucial for solving quadratic equations like the one in the example.