Which are the solutions of the quadratic equation?

x2 = 9x + 6

StartFraction negative 9 minus StartRoot 105 EndRoot Over 2 EndFraction comma StartFraction negative 9 + StartRoot 105 EndRoot Over 2 EndFraction
StartFraction negative 9 minus StartRoot 57 EndRoot Over 2 EndFraction comma StartFraction negative 9 + StartRoot 57 EndRoot Over 2 EndFraction
StartFraction 9 minus StartRoot 105 EndRoot Over 2 EndFraction comma StartFraction 9 + StartRoot 105 EndRoot Over 2 EndFraction
StartFraction 9 minus StartRoot 57 EndRoot Over 2 EndFraction comma StartFraction 9 + StartRoot 57 EndRoot Over 2 EndFraction

1 answer

To solve the quadratic equation \( x^2 = 9x + 6 \), we first rearrange it into the standard form \( ax^2 + bx + c = 0 \).

  1. Subtract \( 9x \) and \( 6 \) from both sides: \[ x^2 - 9x - 6 = 0 \]

  2. Identify the coefficients:

    • \( a = 1 \)
    • \( b = -9 \)
    • \( c = -6 \)
  3. Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the values of \( a \), \( b \), and \( c \): \[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \] \[ x = \frac{9 \pm \sqrt{81 + 24}}{2} \] \[ x = \frac{9 \pm \sqrt{105}}{2} \]

  4. 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.