Problem 17 - Beginning and Intermediate Algebra

Question

Solve using the zero-factor property. $$ 3 x^{2}-13 x=30 $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{5}{3} \) and \( x = 6 \).

Step 1: Move all terms to one side

First, move all terms to one side of the equation to set it equal to zero. Subtract 30 from both sides: \[ 3x^2 - 13x - 30 = 0 \]

Step 2: Factor the quadratic equation

Next, factor the quadratic equation \(3x^2 - 13x - 30\). Look for two numbers that multiply to \(3 \times -30 = -90\) and add up to \(-13\). The numbers are \(-18\) and \(5\) . Using these numbers, we can write: \[ 3x^2 - 18x + 5x - 30 = 0 \] Apply factoring by grouping: \[ 3x(x - 6) + 5(x - 6) = 0 \] Now factor out the common binomial factor: \[ (3x + 5)(x - 6) = 0 \]

Step 3: Apply the zero-factor property

Using the zero-factor property, set each factor equal to zero: \[ 3x + 5 = 0 \] and \[ x - 6 = 0 \] Solve each equation: \[ 3x + 5 = 0 \rightarrow 3x = -5 \rightarrow x = -\frac{5}{3} \] \[ x - 6 = 0 \rightarrow x = 6 \]

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These equations are called 'quadratic' because they involve a variable raised to the second power (the square).

In the given exercise, the quadratic equation begins as \(3x^2 - 13x = 30\). Our first task is to set this equation to zero: \(3x^2 - 13x - 30 = 0\). Setting the equation to zero is essential because it allows us to use various solution methods, such as factoring.

Factoring by Grouping

Factoring by grouping is a method used to simplify polynomials, especially useful for quadratic equations. We want to write the quadratic equation in a way that allows us to factor it into the product of two binomials.

Here's how we do it:

• First, multiply the leading coefficient (3) by the constant term (-30) to get -90.
• Now, find two numbers that multiply to -90 and add up to the middle coefficient (-13). These numbers are -18 and 5.
• Rewriting the middle term using these numbers gives us \(3x^2 - 18x + 5x - 30 = 0\).
• Next, we group the terms: \(3x(x - 6) + 5(x - 6) = 0\).
• Factor out the common binomial factor: \( (3x + 5)(x - 6) = 0 \).

This process simplifies the original quadratic equation into a form that is easier to solve using the zero-factor property.

Solving Algebraic Equations

Now that we have factored the quadratic equation, we can use the zero-factor property to find the solutions. The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero.

For the factors \( (3x + 5)(x - 6) = 0 \), we set each factor to zero and solve for \(x\):

• \(3x + 5 = 0\), solving gives:
  \[ 3x + 5 = 0 \]
  \[ 3x = -5 \]
  \[ x = -\frac{5}{3} \]
• \(x - 6 = 0\), solving gives:
  \[ x - 6 = 0 \]
  \[ x = 6 \]

This means our quadratic equation, \(3x^2 - 13x - 30 \), has two solutions: \(x = -\frac{5}{3}\) and \(x = 6\).

Always check your solutions by substituting them back into the original equation to confirm they satisfy it. This final step ensures that our work is correct.