Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.

Match each equation with its solution set.

[tex]\[
\begin{array}{l}
a^2 - 9a + 14 = 0 \\
a^2 - 5a - 14 = 0
\end{array}
\][/tex]

[tex]\[
a^2 + 9a + 14 = 0 \quad a^2 + 3a - 10 = 0 \quad a^2 + 5a - 14 = 0
\][/tex]

[tex]\[
\{-2, 7\} \\
\{2, -7\} \\
\{-2, -7\} \\
\{7, 2\}
\][/tex]

[tex]\[
\begin{array}{|c|}
\hline
\square \\
\hline
\square \\
\hline
\end{array}
\][/tex]



Answer :

To solve the problem, we need to find the roots of the given quadratic equations and match them with their corresponding solution sets.

Let's begin by solving each quadratic equation step-by-step:

1. Solve [tex]\( a^2 - 9a + 14 = 0 \)[/tex]

This is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].

The solutions [tex]\( a \)[/tex] are given by the quadratic formula:
[tex]\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For [tex]\( a^2 - 9a + 14 = 0 \)[/tex]:
[tex]\[ a = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(14)}}{2(1)} \][/tex]
[tex]\[ a = \frac{9 \pm \sqrt{81 - 56}}{2} \][/tex]
[tex]\[ a = \frac{9 \pm \sqrt{25}}{2} \][/tex]
[tex]\[ a = \frac{9 \pm 5}{2} \][/tex]

So, the solutions are:
[tex]\[ a = \frac{9 + 5}{2} = 7 \quad \text{and} \quad a = \frac{9 - 5}{2} = 2 \][/tex]

Therefore, the solution set for [tex]\( a^2 - 9a + 14 = 0 \)[/tex] is [tex]\( \{7, 2\} \)[/tex].

2. Solve [tex]\( a^2 - 5a - 14 = 0 \)[/tex]

Again using the quadratic formula:
[tex]\[ a = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)} \][/tex]
[tex]\[ a = \frac{5 \pm \sqrt{25 + 56}}{2} \][/tex]
[tex]\[ a = \frac{5 \pm \sqrt{81}}{2} \][/tex]
[tex]\[ a = \frac{5 \pm 9}{2} \][/tex]

So, the solutions are:
[tex]\[ a = \frac{5 + 9}{2} = 7 \quad \text{and} \quad a = \frac{5 - 9}{2} = -2 \][/tex]

Therefore, the solution set for [tex]\( a^2 - 5a - 14 = 0 \)[/tex] is [tex]\( \{7, -2\} \)[/tex].

Now, let's match these solution sets with the given equations.

1. [tex]\( a^2 - 9a + 14 = \{2, 7\} \)[/tex]
2. [tex]\( a^2 - 5a - 14 = \{7, -2\} \)[/tex]

So, the final matching is:

- [tex]\( a^2 - 9a + 14 = 0 \)[/tex] matches with [tex]\( \{2, 7\} \)[/tex]
- [tex]\( a^2 - 5a - 14 = 0 \)[/tex] matches with [tex]\( \{7, -2\} \)[/tex]