https://file.aiquickdraw.com/mathbot/file/1729791868216kex8a147.jpg

The image contains three expressions that need to be factored. Let's go through each of them:

1. **\( x^2 - 15x + 54 \):**
   We need two numbers that multiply to \( 54 \) and add to \( -15 \). The pair that works is \( -6 \) and \( -9 \) because:
   \[
   (-6) \times (-9) = 54 \quad \text{and} \quad (-6) + (-9) = -15
   \]
   Therefore, the factorization is:
   \[
   x^2 - 15x + 54 = (x - 6)(x - 9)
   \]

2. **\( x^2 - 12x - 6 \):**
   Here, we need two numbers that multiply to \( -6 \) and add to \( -12 \). However, there are no integer pairs that satisfy this, so we will need to use the quadratic formula:
   \[
   x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(-6)}}{2(1)} = \frac{12 \pm \sqrt{144 + 24}}{2} = \frac{12 \pm \sqrt{168}}{2}
   \]
   Simplifying \( \sqrt{168} \), we get approximately \( 12.96 \), so:
   \[
   x = \frac{12 \pm 12.96}{2} \quad \text{or approximately} \quad x \approx 12.48 \quad \text{or} \quad x \approx -0.48
   \]
   This means no clean factorization is possible without using decimal numbers or radicals.

3. **\( x^2 + 6x - 16 \):**
   We need two numbers that multiply to \( -16 \) and add to \( 6 \). The pair that works is \( 8 \) and \( -2 \) because:
   \[
   (8) \times (-2) = -16 \quad \text{and} \quad 8 + (-2) = 6
   \]
   Therefore, the factorization is:
   \[
   x^2 + 6x - 16 = (x + 8)(x - 2)
   \]

Would you like more details or explanations on any part?

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Here are five related questions for you:
1. How would you solve a quadratic that does not factor easily?
2. Can you use completing the square to factor non-factorable quadratics?
3. What does the discriminant tell you about the solutions of a quadratic equation?
4. How does factoring quadratics relate to solving equations?
5. Can factoring be used in higher-degree polynomials?

**Tip:** Always check for the greatest common factor (GCF) before trying to factor quadratic expressions!

Factor each expression: x^2 - 15x + 54, x^2 - 12x - 6, x^2 + 6x - 16

Learn how to factor the quadratic equations x^2 - 15x + 54, x^2 - 12x - 6, and x^2 + 6x - 16 with a detailed solution. This problem covers key algebraic concepts such as factoring trinomials and the quadratic formula. Suitable for students in Grades 9-11.

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