To determine which equation in the expanded form matches its factored form, let's analyze each pair:
1. Equation: [tex]\(9 x^2 + 24 x + 17 = 0\)[/tex]
- Factored Form: [tex]\((3x - (-4 + i))(3x + (-4 - i)) = 0\)[/tex]
2. Equation: [tex]\(x^3 - 5x^2 + 11x - 15 = 0\)[/tex]
- Factored Form: [tex]\((x - 3)(x - (1 - 2i))(x - (1 + 2i)) = 0\)[/tex]
3. Equation: [tex]\(6x - 13 = x^2\)[/tex]
- Factored Form: [tex]\((x - (3 + 2i))(x - (-3 + 2i)) = 0\)[/tex]
4. Equation: [tex]\(25x^2 + 36 = 0\)[/tex]
- Factored Form: [tex]\((5x + 6i)(5x + 6i) = 0\)[/tex]
To match each factored form with the expanded form through solving and verification, let's analyze them:
1. Expanding [tex]\((3x - (-4 + i))(3x + (-4 - i))\)[/tex]:
- This expands to [tex]\(9x^2 + 24x + 17\)[/tex]. So, it matches the given expanded form.
2. Expanding [tex]\((x - 3)(x - (1 - 2i))(x - (1 + 2i))\)[/tex]:
- This expands to [tex]\(x^3 - 5x^2 + 11x - 15\)[/tex]. So, it matches the given expanded form.
3. Expanding [tex]\((x - (3 + 2i))(x - (-3 + 2i))\)[/tex]:
- This expands to [tex]\(x^2 - 6x + 13\)[/tex], which does not match [tex]\(6x - 13 = x^2\)[/tex].
4. Expanding [tex]\((5x + 6i)(5x + 6i)\)[/tex]:
- This expands to [tex]\(25x^2 + 36\)[/tex]. So, it matches the given expanded form.
Based on this analysis, the correct expanded form equation that matches its factored form is:
Equation: [tex]\(x^3 - 5x^2 + 11x - 15 = 0\)[/tex]
- Factored Form: [tex]\((x - 3)(x - (1 - 2i))(x - (1 + 2i)) = 0\)[/tex]
Thus, the correct equation is:
[tex]\[ \boxed{2} \][/tex]
