To determine the other solution to the quadratic equation [tex]\(x^2 + 7x + 12 = 0\)[/tex] given that [tex]\(x = -3\)[/tex] is one of the solutions, we can follow these steps:
1. Understand the structure of the quadratic equation:
A quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] typically has two solutions. These solutions can be found using factoring or the quadratic formula. Here, our equation is [tex]\(x^2 + 7x + 12 = 0\)[/tex].
2. Recall that [tex]\(x = -3\)[/tex] is a solution:
Since [tex]\(x = -3\)[/tex] is given as one solution, we recognize that [tex]\(-3\)[/tex] satisfies the equation. This means when we substitute [tex]\(-3\)[/tex] back into the equation, it should equal zero.
3. Factor the quadratic equation:
We can factor [tex]\(x^2 + 7x + 12 = 0\)[/tex] to find its roots. The factors of the quadratic equation are typically in the form [tex]\((x - p)(x - q) = 0\)[/tex] where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are the solutions.
[tex]\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\][/tex]
4. Verify the factored form:
By expanding [tex]\((x + 3)(x + 4)\)[/tex], we get:
[tex]\[
(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12
\][/tex]
This confirms that the factors are correct.
5. Determine the solutions from the factored form:
Setting each factor equal to zero gives the solutions:
[tex]\[
x + 3 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solving these equations, we get:
[tex]\[
x = -3 \quad \text{or} \quad x = -4
\][/tex]
6. Identify the correct solution from the options:
The given options are:
- (A) -5
- (B) -4
- (C) 1
- (D) 2
- (E) 3
From our factored results, we see that [tex]\(x = -4\)[/tex] is the other solution besides [tex]\(x = -3\)[/tex].
Therefore, the other value of [tex]\( x \)[/tex] is [tex]\(\boxed{-4}\)[/tex].
