x2+32x-20=0

To solve the quadratic equation \(x^2 + 32x - 20 = 0\), we can use the quadratic formula:

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

In this equation, \(a = 1\), \(b = 32\), and \(c = -20\).

First, we need to calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 - 4ac = 32^2 - 4 \cdot 1 \cdot (-20) \] \[ = 1024 + 80 \] \[ = 1104 \]

Now, we can substitute back into the quadratic formula:

\[ x = \frac{-32 \pm \sqrt{1104}}{2 \cdot 1} \]

Next, we simplify \(\sqrt{1104}\):

\[ \sqrt{1104} = \sqrt{16 \cdot 69} = 4\sqrt{69} \]

Now substitute this back into the formula:

\[ x = \frac{-32 \pm 4\sqrt{69}}{2} \]

Simplifying this gives:

\[ x = \frac{-32}{2} \pm \frac{4\sqrt{69}}{2} \] \[ x = -16 \pm 2\sqrt{69} \]

Thus, the solutions to the equation \(x^2 + 32x - 20 = 0\) are:

\[ x = -16 + 2\sqrt{69} \quad \text{and} \quad x = -16 - 2\sqrt{69} \]