Sure, let's solve the quadratic equation [tex]\(x^2 + 5x + 3 = 0\)[/tex] step by step.
1. Identify the coefficients:
- The equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
- Here, [tex]\(a = 1\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = 3\)[/tex].
2. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values, we get:
[tex]\[
\Delta = 5^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13
\][/tex]
3. Apply the quadratic formula:
The solutions of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] are given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting [tex]\(a = 1\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(\Delta = 13\)[/tex]:
[tex]\[
x = \frac{-5 \pm \sqrt{13}}{2}
\][/tex]
4. Find the two solutions:
- For the [tex]\(+\)[/tex] case:
[tex]\[
x_1 = \frac{-5 + \sqrt{13}}{2}
\][/tex]
- For the [tex]\(-\)[/tex] case:
[tex]\[
x_2 = \frac{-5 - \sqrt{13}}{2}
\][/tex]
Therefore, the solutions to the equation [tex]\(x^2 + 5x + 3 = 0\)[/tex] are:
[tex]\[
x_1 = \frac{-5 - \sqrt{13}}{2} \quad \text{and} \quad x_2 = \frac{-5 + \sqrt{13}}{2}
\][/tex]
So, the solutions are [tex]\(\boxed{\frac{-5 - \sqrt{13}}{2}}\)[/tex] and [tex]\(\boxed{\frac{-5 + \sqrt{13}}{2}}\)[/tex].
