Use factoring to solve the quadratic equation.  Check by substitution or by using a graphing utility and identifying​ x-intercepts.
x squared minus 3 x minus 28 equals 0
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Part 1
The solution set is StartSet nothing EndSet
  
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​(Use a comma to separate answers as needed. Type repeated roots only​ once.)


To solve the quadratic equation \( x^2 - 3x - 28 = 0 \) by factoring, follow these steps:

### Step 1: Factor the quadratic equation
We need to find two numbers that multiply to give the constant term \(-28\) and add up to the coefficient of the middle term, \(-3\).

- The two numbers are \( -7 \) and \( 4 \), because:
  \[
  -7 \times 4 = -28 \quad \text{and} \quad -7 + 4 = -3
  \]

Thus, we can factor the quadratic expression as:
\[
x^2 - 3x - 28 = (x - 7)(x + 4) = 0
\]

### Step 2: Solve for \(x\)
Set each factor equal to zero:
\[
x - 7 = 0 \quad \text{or} \quad x + 4 = 0
\]

Solving these gives:
\[
x = 7 \quad \text{or} \quad x = -4
\]

### Solution Set:
The solution set is:
\[
\{ 7, -4 \}
\]

### Step 3: Check by substitution
To verify, substitute \(x = 7\) and \(x = -4\) into the original equation:
- For \(x = 7\):
  \[
  7^2 - 3(7) - 28 = 49 - 21 - 28 = 0
  \]
  Thus, \(x = 7\) is a solution.

- For \(x = -4\):
  \[
  (-4)^2 - 3(-4) - 28 = 16 + 12 - 28 = 0
  \]
  Thus, \(x = -4\) is a solution.

Both solutions are correct.

### Final Answer:
The solution set is:
\[
\boxed{7, -4}
\]

Would you like to check using a graphing utility or ask for further details?

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Here are five related questions to expand your understanding:
1. How do you solve quadratic equations using the quadratic formula?
2. What is the relationship between the discriminant and the nature of the roots?
3. How can you solve a quadratic equation by completing the square?
4. How does factoring relate to the x-intercepts of a quadratic graph?
5. What happens when a quadratic equation has repeated roots?

**Tip:** When factoring, always check if the quadratic can be simplified first, such as factoring out a common factor.

Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying x-intercepts. x squared minus 3x minus 28 equals 0.

Learn how to solve the quadratic equation x^2 - 3x - 28 by factoring, with a step-by-step solution and verification using substitution. This tutorial covers key algebraic concepts, including factoring and the zero product property, and is suitable for Grades 8-10.

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