Certainly! Let's go through and solve each of the given equations by factoring. We'll then enter both the factors and their solutions in the table.
1. Equation: [tex]\(x^2 + 10 = 0\)[/tex]
- To solve for [tex]\(x\)[/tex], we need to factor the equation. However, [tex]\(x^2 + 10\)[/tex] doesn't factor nicely into real numbers. Instead, we solve it directly.
- Solve [tex]\(x^2 + 10 = 0\)[/tex]:
- [tex]\(x^2 = -10\)[/tex]
- [tex]\(x = \pm \sqrt{-10}\)[/tex]
- Since [tex]\(\sqrt{-10} = \sqrt{10} \cdot i\)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit), the solutions are [tex]\(x = \pm \sqrt{10} \cdot i\)[/tex].
- Thus, the solutions are [tex]\(x = -\sqrt{10} \cdot i\)[/tex] and [tex]\(x = \sqrt{10} \cdot i\)[/tex].
2. Equation: [tex]\(4x^2 + 25 = 0\)[/tex]
- To solve, we set the equation to zero:
- [tex]\(4x^2 + 25 = 0\)[/tex]
- [tex]\(4x^2 = -25\)[/tex]
- [tex]\(x^2 = -\frac{25}{4}\)[/tex]
- [tex]\(x = \pm \sqrt{-\frac{25}{4}}\)[/tex]
- Since [tex]\(\sqrt{-\frac{25}{4}} = \frac{\sqrt{25}}{2} \cdot i = \frac{5i}{2}\)[/tex], the solutions are [tex]\(x = \pm \frac{5i}{2}\)[/tex].
- Thus, the solutions are [tex]\(x = -\frac{5i}{2}\)[/tex] and [tex]\(x = \frac{5i}{2}\)[/tex].
3. Equation: [tex]\(x^2 + 121 = 0\)[/tex]
- To solve, we set the equation to zero:
- [tex]\(x^2 + 121 = 0\)[/tex]
- [tex]\(x^2 = -121\)[/tex]
- [tex]\(x = \pm \sqrt{-121}\)[/tex]
- Since [tex]\(\sqrt{-121} = \sqrt{121} \cdot i = 11i\)[/tex], the solutions are [tex]\(x = \pm 11i\)[/tex].
- Thus, the solutions are [tex]\(x = -11i\)[/tex] and [tex]\(x = 11i\)[/tex].
Now, we'll fill in the table with the factors (which, in these cases, are the same as the original equations since they can't be factored into simpler real-number terms) and the solutions:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline Equation & Factors & Solutions \\
\hline$x^2+10=0$ & $x^2 + 10$ & $-\sqrt{10}i, \sqrt{10}i$ \\
\hline $4 x^2+25=0$ & $4x^2 + 25$ & $-\frac{5i}{2}, \frac{5i}{2}$ \\
\hline$x^2+121=0$ & $x^2 + 121$ & $-11i, 11i$ \\
\hline
\end{tabular}
\][/tex]
This table now correctly contains the factors and the solutions for each quadratic equation.
