To solve the quadratic equation [tex]\(x^2 + 4x - 21 = 0\)[/tex], we follow these steps:
1. Identify the coefficients: The equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -21\)[/tex].
2. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] is given by the formula [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
\Delta = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-21) = 16 + 84 = 100
\][/tex]
3. Determine the roots using the quadratic formula: The solutions to the equation are found using:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(\Delta = 100\)[/tex], we get:
[tex]\[
x = \frac{-4 \pm \sqrt{100}}{2 \cdot 1} = \frac{-4 \pm 10}{2}
\][/tex]
This results in two solutions:
[tex]\[
x_1 = \frac{-4 + 10}{2} = \frac{6}{2} = 3
\][/tex]
[tex]\[
x_2 = \frac{-4 - 10}{2} = \frac{-14}{2} = -7
\][/tex]
4. Verify the solutions: Compare the calculated solutions with the given options:
- Option A: [tex]\(x = -7\)[/tex] matches solution [tex]\(x_2 = -7\)[/tex].
- Option B: [tex]\(x = 3\)[/tex] matches solution [tex]\(x_1 = 3\)[/tex].
- Option C: [tex]\(x = 4\)[/tex] does not match either solution.
- Option D: [tex]\(x = 21\)[/tex] does not match either solution.
- Option E: [tex]\(x = -5\)[/tex] does not match either solution.
Thus, the correct solutions are:
A. [tex]\(x = -7\)[/tex]
B. [tex]\(x = 3\)[/tex]
Therefore, options A and B are correct.
