https://file.aiquickdraw.com/mathbot/file/1726798443157orvv78v4.jpg

The image contains two math problems:

1. **Problem 1: Solve for \( x \)**
   \[
   P(x) = x^3 - 8x^2 + x + 42 = 0
   \]
   With the multiple-choice answers provided as:
   - \( -7, -2, 3 \)
   - \( -2, 3, 7 \)
   - \( 2, 3, 7 \)
   - \( -3, -2, 7 \)
   - None of these

2. **Problem 2: Solve for the exact values of \( x \)**
   \[
   x^3 + 4x^2 - 12x = -7
   \]
   With multiple-choice answers formatted in terms of radicals.

I'll start by solving both equations step by step. Let's begin with the first equation.For the first problem, the solutions to the equation \( P(x) = x^3 - 8x^2 + x + 42 = 0 \) are \( x = -2, 3, 7 \).

This matches the multiple-choice answer \( -2, 3, 7 \).

Next, I'll solve the second equation \( x^3 + 4x^2 - 12x = -7 \).For the second problem, the solutions to the equation \( x^3 + 4x^2 - 12x = -7 \) are:

1. \( x = 1 \)
2. \( x = \frac{-5 + \sqrt{53}}{2} \)
3. \( x = \frac{-5 - \sqrt{53}}{2} \)

This matches the multiple-choice answer:
\[
1, \frac{-5 \pm \sqrt{53}}{2}
\]

Would you like a detailed explanation of any of the steps? Here are 5 questions to further explore these problems:

1. What method is used to solve cubic equations?
2. How do you check if a solution to a cubic equation is correct?
3. What does the \(\pm\) symbol indicate in the solutions for radicals?
4. How can you graph the functions \( P(x) \) and \( x^3 + 4x^2 - 12x + 7 \) to verify the roots visually?
5. What is the significance of rational and irrational solutions in these equations?

**Tip:** Always verify your solutions by substituting them back into the original equation!

Solve the cubic equations: 1. P(x) = x^3 - 8x^2 + x + 42 = 0 and 2. x^3 + 4x^2 - 12x = -7.

Learn how to solve two cubic equations: x^3 - 8x^2 + x + 42 = 0 and x^3 + 4x^2 - 12x = -7. This solution covers key concepts such as factoring, radical solutions, and applying the Rational Root Theorem. Ideal for high school students in Grades 10-12.

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