Solve the equation by factoring.

x squared equals 4 x plus 21x2=4x+21

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Part 1

The solution set is 

StartSet nothing EndSetenter your response here.

​(Use a comma to separate answers as needed. Type each solution only​ once.)

To solve the quadratic equation \( x^2 = 4x + 21 \) by factoring, follow these steps:

### Step 1: Move all terms to one side of the equation
First, subtract \( 4x \) and \( 21 \) from both sides to set the equation to zero:
\[
x^2 - 4x - 21 = 0
\]

### Step 2: Factor the quadratic expression
We need to factor the quadratic expression \( x^2 - 4x - 21 \). Look for two numbers that multiply to \(-21\) (the constant term) and add to \(-4\) (the coefficient of \(x\)).

The numbers that work are \(-7\) and \(3\), because:
\[
-7 \times 3 = -21 \quad \text{and} \quad -7 + 3 = -4
\]

So, we can factor the quadratic as:
\[
(x - 7)(x + 3) = 0
\]

### Step 3: Solve for \(x\) using the Zero Product Property
Set each factor equal to zero and solve for \(x\):
1. \( x - 7 = 0 \) gives \( x = 7 \)
2. \( x + 3 = 0 \) gives \( x = -3 \)

### Step 4: Write the solution set
The solution set is \( \{7, -3\} \).

### Final Answer
The solution set is \( \{7, -3\} \).

Would you like more details or have any questions about this solution?

### Follow-up Questions:
1. How can you check if the solutions you found are correct?
2. What happens if the quadratic cannot be factored easily?
3. How would you solve the same equation using the quadratic formula?
4. What are the possible shapes of the graph of a quadratic function?
5. Can a quadratic equation have complex solutions? When does this happen?

**Tip:** When factoring, always check your factors by expanding them to ensure they match the original quadratic expression.

Learn how to solve the quadratic equation x^2 = 4x + 21 step-by-step using factoring. Understand the process of moving terms, factoring quadratic expressions, and applying the Zero Product Property. Suitable for students in Grades 7-9.

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