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Match each equation with its solution set.
[tex]\[
\begin{array}{l}
a^2 - 9a + 14 = 0 \quad \{2, 7\} \\
a^2 + 9a + 14 = 0 \quad \{-2, -7\} \\
a^2 + 3a - 10 = 0 \quad \{-5, 2\} \\
a^2 + 5a - 14 = 0 \quad \{-7, 2\} \\
a^2 - 5a - 14 = 0 \quad \{7, -2\} \\
\end{array}
\][/tex]

Sagot :

GinnyAnswer
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]
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