Let's solve each quadratic equation step by step to find the correct solution sets.
1. Equation: [tex]\(a^2 - 9a + 14 = 0\)[/tex]
To solve this, we can factorize the quadratic expression:
[tex]\[
a^2 - 9a + 14 = (a - 7)(a - 2) = 0
\][/tex]
Setting each factor to zero gives us:
[tex]\[
a - 7 = 0 \quad \text{or} \quad a - 2 = 0
\][/tex]
Thus, the solutions are:
[tex]\[
a = 7 \quad \text{and} \quad a = 2
\][/tex]
Solution set: [tex]\(\{7, 2\}\)[/tex]
2. Equation: [tex]\(a^2 + 9a + 14 = 0\)[/tex]
To solve this, we can factorize the quadratic expression:
[tex]\[
a^2 + 9a + 14 = (a + 7)(a + 2) = 0
\][/tex]
Setting each factor to zero gives us:
[tex]\[
a + 7 = 0 \quad \text{or} \quad a + 2 = 0
\][/tex]
Thus, the solutions are:
[tex]\[
a = -7 \quad \text{and} \quad a = -2
\][/tex]
Solution set: [tex]\(\{-7, -2\}\)[/tex]
3. Equation: [tex]\(a^2 + 3a - 10 = 0\)[/tex]
To solve this, we can factorize the quadratic expression:
[tex]\[
a^2 + 3a - 10 = (a + 5)(a - 2) = 0
\][/tex]
Setting each factor to zero gives us:
[tex]\[
a + 5 = 0 \quad \text{or} \quad a - 2 = 0
\][/tex]
Thus, the solutions are:
[tex]\[
a = -5 \quad \text{and} \quad a = 2
\][/tex]
Solution set: [tex]\(\{2, -5\}\)[/tex]
4. Equation: [tex]\(a^2 + 5a - 14 = 0\)[/tex]
To solve this, we can factorize the quadratic expression:
[tex]\[
a^2 + 5a - 14 = (a + 7)(a - 2) = 0
\][/tex]
Setting each factor to zero gives us:
[tex]\[
a + 7 = 0 \quad \text{or} \quad a - 2 = 0
\][/tex]
Thus, the solutions are:
[tex]\[
a = -4 \quad \text{and} \quad a = 2
\][/tex]
Solution set: [tex]\(\{2, -4\}\)[/tex]
5. Equation: [tex]\(a^2 - 5a - 14 = 0\)[/tex]
To solve this, we can factorize the quadratic expression:
[tex]\[
a^2 - 5a - 14 = (a - 7)(a + 2) = 0
\][/tex]
Setting each factor to zero gives us:
[tex]\[
a - 7 = 0 \quad \text{or} \quad a + 2 = 0
\][/tex]
Thus, the solutions are:
[tex]\[
a = 7 \quad \text{and} \quad a = -2
\][/tex]
Solution set: [tex]\(\{7, -2\}\)[/tex]
So, the correct matches are:
1. [tex]\(a^2 - 9a + 14 = 0 \quad \longrightarrow \quad \{7, 2\}\)[/tex]
2. [tex]\(a^2 + 9a + 14 = 0 \quad \longrightarrow \quad \{-7, -2\}\)[/tex]
3. [tex]\(a^2 + 3a - 10 = 0 \quad \longrightarrow \quad \{2, -5\}\)[/tex]
4. [tex]\(a^2 + 5a - 14 = 0 \quad \longrightarrow \quad \{2, -4\}\)[/tex]
5. [tex]\(a^2 - 5a - 14 = 0 \quad \longrightarrow \quad \{7, -2\}\)[/tex]
