To solve the quadratic equation:
[tex]\[ x^2 = 9x + 6 \][/tex]
we start by rewriting it in the standard quadratic form:
[tex]\[ x^2 - 9x - 6 = 0 \][/tex]
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), we solve it using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, \(a = 1\), \(b = -9\), and \(c = -6\). Substituting these values into the quadratic formula gives:
[tex]\[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{81 + 24}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{105}}{2} \][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[ x = \frac{9 - \sqrt{105}}{2} \][/tex]
[tex]\[ x = \frac{9 + \sqrt{105}}{2} \][/tex]
Comparing these with the given options, the correct answer is:
[tex]\[ \frac{9-\sqrt{105}}{2}, \frac{9+\sqrt{105}}{2} \][/tex]
