To identify the solutions to the quadratic equation [tex]\(x^2 + 3x - 28 = 0\)[/tex], we can proceed as follows:
1. Set up the equation:
[tex]\[
x^2 + 3x - 28 = 0
\][/tex]
2. Factor the quadratic expression:
To factor the quadratic expression, find two numbers that multiply to [tex]\(-28\)[/tex] (the constant term) and add up to [tex]\(3\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
These two numbers are [tex]\(7\)[/tex] and [tex]\(-4\)[/tex]:
[tex]\[
x^2 + 3x - 28 = (x + 7)(x - 4)
\][/tex]
3. Set each factor equal to zero:
Once factored, set each factor equal to zero to find the roots (solutions):
[tex]\[
(x + 7) = 0 \quad \text{or} \quad (x - 4) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 7 = 0 \implies x = -7
\][/tex]
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(x^2 + 3x - 28 = 0\)[/tex] are [tex]\(x = -7\)[/tex] and [tex]\(x = 4\)[/tex].
Now, let's check which of the given options are solutions:
- Option A: [tex]\(x = -7\)[/tex]: This is one of the solutions we found. So, it is correct.
- Option B: [tex]\(x = 28\)[/tex]: This is not one of the solutions we found. So, it is incorrect.
- Option C: [tex]\(x = 4\)[/tex]: This is the other solution we found. So, it is correct.
- Option D: [tex]\(x = -5\)[/tex]: This is not one of the solutions we found. So, it is incorrect.
- Option E: [tex]\(x = 3\)[/tex]: This is not one of the solutions we found. So, it is incorrect.
Therefore, the correct options are:
A. [tex]\(x = -7\)[/tex]
C. [tex]\(x = 4\)[/tex]
