Which are the solutions of the quadratic equation?

x2 = 7x + 4

StartFraction negative 7 minus StartRoot 65 EndRoot Over 2 EndFraction comma StartFraction negative 7 + StartRoot 65 EndRoot Over 2 EndFraction
–7, 0
StartFraction 7 minus StartRoot 65 EndRoot Over 2 EndFraction comma StartFraction 7 + StartRoot 65 EndRoot Over 2 EndFraction
7, 0

1 answer

To find the solutions of the quadratic equation \( x^2 = 7x + 4 \), we first rearrange it into standard form:

\[ x^2 - 7x - 4 = 0 \]

Now, we can use the quadratic formula to solve for \( x \), which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, \( a = 1 \), \( b = -7 \), and \( c = -4 \). Plugging in these values:

  1. Calculate the discriminant: \[ b^2 - 4ac = (-7)^2 - 4(1)(-4) = 49 + 16 = 65 \]

  2. Plug the values into the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{65}}{2(1)} = \frac{7 \pm \sqrt{65}}{2} \]

Thus, the solutions are:

\[ x = \frac{7 - \sqrt{65}}{2} \quad \text{and} \quad x = \frac{7 + \sqrt{65}}{2} \]

This corresponds to the option:

\[ \frac{7 - \sqrt{65}}{2}, \frac{7 + \sqrt{65}}{2} \]

So the final answer is:

\(\frac{7 - \sqrt{65}}{2}, \frac{7 + \sqrt{65}}{2}\)