To solve the quadratic equation [tex]\( x^2 + 3x - 4 = 6 \)[/tex], follow these steps:
1. Move all terms to one side of the equation to set it to zero:
[tex]\[
x^2 + 3x - 4 - 6 = 0
\][/tex]
Simplify:
[tex]\[
x^2 + 3x - 10 = 0
\][/tex]
2. Factor the quadratic equation [tex]\(x^2 + 3x - 10 = 0\)[/tex]:
To factor the quadratic expression, we need to find two numbers that multiply to [tex]\(-10\)[/tex] (the constant term) and add up to [tex]\(3\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term).
The numbers that satisfy these conditions are [tex]\(5\)[/tex] and [tex]\(-2\)[/tex] because:
[tex]\[
5 \times (-2) = -10 \quad \text{and} \quad 5 + (-2) = 3
\][/tex]
So, we can factor the equation as:
[tex]\[
(x + 5)(x - 2) = 0
\][/tex]
3. Solve the factored equation by setting each factor to zero:
[tex]\[
x + 5 = 0 \quad \text{or} \quad x - 2 = 0
\][/tex]
Solving these:
[tex]\[
x = -5 \quad \text{or} \quad x = 2
\][/tex]
4. Determine the solution set:
The solution set is:
[tex]\[
\{-5, 2\}
\][/tex]
Considering the solution set, we match it with the given choices:
A. [tex]\(\{-5, 2\}\)[/tex]
B. [tex]\(\{-2, -1\}\)[/tex]
C. [tex]\(\{2, 7\}\)[/tex]
D. [tex]\(\{5, 10\}\)[/tex]
The correct option is:
[tex]\(\boxed{A}\)[/tex]
