Question

which are the solutions of the quadratic equation?
x² = 9x + 6
\\(\frac{-9 - \sqrt{105}}{2}\\), \\(\frac{-9 + \sqrt{105}}{2}\\)
\\(\frac{-9 - \sqrt{57}}{2}\\), \\(\frac{-9 + \sqrt{57}}{2}\\)
\\(\frac{9 - \sqrt{57}}{2}\\), \\(\frac{9 + \sqrt{57}}{2}\\)

Explanation:

Step1: Rearrange to standard form

$x^2 - 9x - 6 = 0$

Step2: Identify coefficients

For $ax^2+bx+c=0$, $a=1$, $b=-9$, $c=-6$

Step3: Apply quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$x=\frac{-(-9)\pm\sqrt{(-9)^2-4(1)(-6)}}{2(1)}$

Step4: Simplify the expression

Calculate discriminant: $(-9)^2-4(1)(-6)=81+24=105$
$x=\frac{9\pm\sqrt{105}}{2}$
Wait, correction: Recheck discriminant calculation error in options match. Wait no, original equation $x^2=9x+6$ → $x^2-9x-6=0$. Discriminant is $81 +24=105$. Wait, no option matches? Wait no, recheck options:
Wait, no, I misread options. Wait, no, let's re-express:
Wait, $\frac{9\pm\sqrt{105}}{2}$ is same as $\frac{-(-9)\pm\sqrt{105}}{2}$. Wait, the first option is $\frac{-9\pm\sqrt{105}}{2}$, which is wrong. Wait no, wait my mistake: no, $b$ is $-9$, so $-b$ is $9$. Wait, but let's recalculate:
Wait $x^2 -9x -6=0$
$x=\frac{9\pm\sqrt{81 +24}}{2}=\frac{9\pm\sqrt{105}}{2}$. But this is not listed? Wait no, wait the user's image: wait the first option is $\frac{-9-\sqrt{105}}{2},\frac{-9+\sqrt{105}}{2}$ – no, that would be if equation was $x^2+9x+6=0$. Wait, did I rearrange wrong?
Original equation: $x^2=9x+6$ → $x^2 -9x -6=0$. Correct.
Wait, maybe a typo in options? No, wait wait, maybe I miscalculated discriminant: $b^2-4ac = (-9)^2 -4*1*(-6)=81+24=105$. Correct.
Wait, but none of the options have $\sqrt{105}$ with positive 9. Wait, the first option has $\sqrt{105}$ with -9. That would be for $x^2+9x+6=0$.
Wait, maybe the user's image has a typo? Or did I misread the original equation?
Wait original equation: $x^2=9x+6$. Yes.
Wait, wait, maybe the options are written as $\frac{9\pm\sqrt{105}}{2}$ but displayed as $\frac{-9\pm\sqrt{105}}{2}$? No, no. Wait no, wait let's check again:
Wait, no, if the equation was $x^2=-9x+6$, then $x^2+9x-6=0$, discriminant $81+24=105$, solutions $\frac{-9\pm\sqrt{105}}{2}$, which is first option. But original equation is $x^2=9x+6$.
Wait, maybe I made a mistake in sign. Wait quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. For $x^2-9x-6=0$, $b=-9$, so $-b=9$. Correct.
Wait, but the options given: first option is $\frac{-9-\sqrt{105}}{2},\frac{-9+\sqrt{105}}{2}$, second is $\frac{-9\pm\sqrt{57}}{2}$, third is $\frac{9\pm\sqrt{57}}{2}$.
Ah! Wait, discriminant for $x^2-9x+6=0$ would be $81-24=57$, solutions $\frac{9\pm\sqrt{57}}{2}$, which is third option. Oh! Maybe the original equation was $x^2=9x-6$? But user wrote $x^2=9x+6$.
Wait, let's recheck the original image: the user's image says $x^2=9x+6$.
Wait, this is a contradiction. Wait, no, wait $x^2=9x+6$ → $x^2-9x-6=0$, discriminant 105, solutions $\frac{9\pm\sqrt{105}}{2}$, which is not listed. But the first option is $\frac{-9\pm\sqrt{105}}{2}$, which is for $x^2+9x+6=0$.
Wait, maybe a typo in the question? Or maybe I misread the sign. Wait, no, let's proceed with the correct calculation for the given equation.

Wait, no, wait maybe the user's image has a typo, but let's follow the correct steps for the given equation $x^2=9x+6$.

Answer:

The correct solutions are $\frac{9+\sqrt{105}}{2}$ and $\frac{9-\sqrt{105}}{2}$. Among the provided options, there is a discrepancy, but if we assume a sign error in the question (equation as $x^2=-9x+6$), the first option would be correct. However, for the given equation $x^2=9x+6$, the solutions are $\frac{9\pm\sqrt{105}}{2}$.

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