What is the product of the four solutions to the equation |x2+x−9|=3
?

72

24

−24

−72

To find the product of the four solutions to the equation \( |x^2 + x - 9| = 3 \), we can approach this by considering the two possible cases due to the absolute value:

1. \( x^2 + x - 9 = 3 \)
2. \( x^2 + x - 9 = -3 \)

### Case 1: \( x^2 + x - 9 = 3 \)

Simplify and solve the equation:
\[
x^2 + x - 12 = 0
\]
Factor the quadratic equation:
\[
(x - 3)(x + 4) = 0
\]
So, the solutions are:
\[
x = 3 \quad \text{or} \quad x = -4
\]

### Case 2: \( x^2 + x - 9 = -3 \)

Simplify and solve the equation:
\[
x^2 + x - 6 = 0
\]
Factor the quadratic equation:
\[
(x - 2)(x + 3) = 0
\]
So, the solutions are:
\[
x = 2 \quad \text{or} \quad x = -3
\]

### Product of All Solutions

The four solutions we found are \( x = 3, -4, 2, -3 \).

The product of these solutions is:
\[
3 \times (-4) \times 2 \times (-3) = 72
\]

So, the correct answer is **72**.

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Would you like more details, or do you have any other questions? 

Here are five related questions:
1. How do you factor quadratic equations?
2. What are the properties of absolute value functions?
3. Can you explain the significance of the product of the roots in a quadratic equation?
4. How would the solution change if the equation were \( |x^2 + x - 9| = 0 \)?
5. How can you solve higher-degree polynomial equations?

**Tip:** When solving equations involving absolute values, always remember to consider both the positive and negative cases.

Discover how to find the product of all solutions to the equation |x^2 + x - 9| = 3. This problem involves solving quadratic equations with absolute values, factoring, and calculating products of roots. Suitable for Grades 9-12.

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