Which are the solutions of x2 = 19x + 1?

StartFraction 19 minus StartRoot 19 EndRoot Over 2 EndFraction comma StartFraction 19 + StartRoot 19 EndRoot Over 2 EndFraction
StartFraction 19 minus StartRoot 365 EndRoot Over 2 EndFraction comma StartFraction 19 + StartRoot 365 EndRoot Over 2 EndFraction
StartFraction negative 19 minus StartRoot 19 EndRoot Over 2 EndFraction comma StartFraction negative 19 + StartRoot 19 EndRoot Over 2 EndFraction
StartFraction negative 19 minus StartRoot 365 EndRoot Over 2 EndFraction comma StartFraction negative 19 + StartRoot 365 EndRoot Over 2 EndFraction

1 answer

To solve the equation \(x^2 = 19x + 1\), we first rearrange it into standard form:

\[ x^2 - 19x - 1 = 0. \]

Next, we apply the quadratic formula:

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

where \(a = 1\), \(b = -19\), and \(c = -1\). Plugging in these values gives:

\[ x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}. \]

Now, calculating the components:

  1. Calculate \(b^2\):

\[ (-19)^2 = 361. \]

  1. Calculate \(4ac\):

\[ 4 \cdot 1 \cdot (-1) = -4. \]

  1. Now compute \(b^2 - 4ac\):

\[ 361 - (-4) = 361 + 4 = 365. \]

Now, substituting back into the formula:

\[ x = \frac{19 \pm \sqrt{365}}{2}. \]

Thus, the solutions to the equation are:

\[ x = \frac{19 - \sqrt{365}}{2} \quad \text{and} \quad x = \frac{19 + \sqrt{365}}{2}. \]

Given the options, the appropriate pair of solutions is:

\[ \frac{19 - \sqrt{365}}{2}, \frac{19 + \sqrt{365}}{2}. \]

So the correct answer is:

\[ \text{StartFraction } 19 - \text{StartRoot } 365 \text{ EndRoot} \text{ Over } 2 \text{ EndFraction, \text{ StartFraction } 19 + \text{StartRoot } 365 \text{ EndRoot} \text{ Over } 2 \text{ EndFraction}. \]