Sure, let's solve the quadratic equation [tex]\(x^2 + 7x + 4 = 0\)[/tex] using the quadratic formula.
The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Here, the coefficients are [tex]\(a = 1\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 4\)[/tex].
1. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = 7^2 - 4 \cdot 1 \cdot 4 = 49 - 16 = 33
\][/tex]
2. Apply the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{\text{Discriminant}}}}}{2a} = \frac{{-7 \pm \sqrt{33}}}{2 \cdot 1} = \frac{{-7 \pm \sqrt{33}}}{2}
\][/tex]
Now, let's find the two solutions by separately considering the `plus` and `minus` parts of the formula:
- First solution ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{{-7 + \sqrt{33}}}{2}
\][/tex]
- Second solution ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{{-7 - \sqrt{33}}}{2}
\][/tex]
Putting it all together, the solutions to the quadratic equation [tex]\(x^2 + 7x + 4 = 0\)[/tex] are:
[tex]\[
x_1 = \frac{{-7 + \sqrt{33}}}{2} \quad \text{and} \quad x_2 = \frac{{-7 - \sqrt{33}}}{2}
\][/tex]
These solutions are approximately [tex]\(-0.628\)[/tex] and [tex]\(-6.372\)[/tex], respectively.
So, the correct answer from the provided options is:
[tex]\[
x = \frac{{-7 \pm \sqrt{33}}}{2}
\][/tex]
