To determine the exact solutions of the quadratic equation [tex]\(x^2 - 5x - 7 = 0\)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \][/tex]
Here's the step-by-step process:
1. Identify the coefficients from the equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = -7\)[/tex]
2. Calculate the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = (-5)^2 - 4 \cdot 1 \cdot (-7) = 25 + 28 = 53
\][/tex]
3. Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-5) \pm \sqrt{53}}{2 \cdot 1} = \frac{5 \pm \sqrt{53}}{2}
\][/tex]
Therefore, the exact solutions of the quadratic equation [tex]\(x^2 - 5x - 7 = 0\)[/tex] are:
[tex]\[
x = \frac{5 + \sqrt{53}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{53}}{2}
\][/tex]
Given the options:
1. [tex]\(x = \frac{-5 \pm \sqrt{3}}{2}\)[/tex]
2. [tex]\(x = \frac{5 \pm \sqrt{3}}{2}\)[/tex]
3. [tex]\(x = \frac{-5 \pm \sqrt{53}}{2}\)[/tex]
4. [tex]\(x = \frac{5 \pm \sqrt{53}}{2}\)[/tex]
The correct answer matches option 4:
[tex]\[
x = \frac{5 \pm \sqrt{53}}{2}
\][/tex]
