To find the solutions of the quadratic equation [tex]\(x^2 - 9x + 5 = 0\)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
[tex]\[ a = 1, \quad b = -9, \quad c = 5 \][/tex]
Step-by-Step Solution:
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-9)^2 - 4(1)(5) \][/tex]
[tex]\[ \Delta = 81 - 20 \][/tex]
[tex]\[ \Delta = 61 \][/tex]
2. Substitute the values into the quadratic formula:
[tex]\[ x = \frac{-(-9) \pm \sqrt{61}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{61}}{2} \][/tex]
Thus, the solutions are:
[tex]\[ x_1 = \frac{9 + \sqrt{61}}{2} \][/tex]
[tex]\[ x_2 = \frac{9 - \sqrt{61}}{2} \][/tex]
3. Approximate the solutions numerically for clarity:
The approximate numerical values of the solutions are:
[tex]\[ x_1 \approx 8.405124837953327 \][/tex]
[tex]\[ x_2 \approx 0.594875162046673 \][/tex]
Therefore, the exact solutions of the quadratic equation [tex]\(x^2 - 9x + 5 = 0\)[/tex] are:
[tex]\[ x = \frac{9 + \sqrt{61}}{2} \][/tex]
and
[tex]\[ x = \frac{9 - \sqrt{61}}{2} \][/tex]
With the discriminant being 61, which confirms the solutions are real and distinct.
