To solve the quadratic equation \( x^2 = 9x + 6 \), we first rearrange it into the standard form \( ax^2 + bx + c = 0 \).
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Subtract \( 9x \) and \( 6 \) from both sides: \[ x^2 - 9x - 6 = 0 \]
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Identify the coefficients:
- \( a = 1 \)
- \( b = -9 \)
- \( c = -6 \)
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Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the values of \( a \), \( b \), and \( c \): \[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \] \[ x = \frac{9 \pm \sqrt{81 + 24}}{2} \] \[ x = \frac{9 \pm \sqrt{105}}{2} \]
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Thus, the solutions of the equation are: \[ x = \frac{9 - \sqrt{105}}{2} \quad \text{and} \quad x = \frac{9 + \sqrt{105}}{2} \]
Looking at your provided choices, the correct answer is: \[ \text{StartFraction 9 minus StartRoot 105 EndRoot Over 2 EndFraction, StartFraction 9 plus StartRoot 105 EndRoot Over 2 EndFraction} \]
Therefore, the correct option is: StartFraction 9 minus StartRoot 105 EndRoot Over 2 EndFraction, StartFraction 9 plus StartRoot 105 EndRoot Over 2 EndFraction.