To solve the equation \( x^2 - 10 = 30x \), we can first rearrange it into standard quadratic form:
\[ x^2 - 30x - 10 = 0 \]
Now, we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 1 \), \( b = -30 \), and \( c = -10 \). Plugging these values into the formula gives us:
\[ x = \frac{-(-30) \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \]
Calculating \( b^2 - 4ac \):
\[ (-30)^2 = 900 \] \[ 4 \cdot 1 \cdot (-10) = -40 \] \[ b^2 - 4ac = 900 + 40 = 940 \]
Now, substituting this back into the quadratic formula:
\[ x = \frac{30 \pm \sqrt{940}}{2} \]
We simplify \( \sqrt{940} \):
\[ \sqrt{940} = \sqrt{4 \cdot 235} = 2\sqrt{235} \]
Now substitute back:
\[ x = \frac{30 \pm 2\sqrt{235}}{2} \]
This simplifies to:
\[ x = 15 \pm \sqrt{235} \]
Thus, the solutions are:
\[ x = 15 - \sqrt{235} \quad \text{and} \quad x = 15 + \sqrt{235} \]
From your options, the correct solution set is:
\[ { 15 - \sqrt{235}, 15 + \sqrt{235} } \]