X^2+32x-20=0

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

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

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

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

\[ b^2 = 32^2 = 1024 \]

\[ 4ac = 4 \cdot 1 \cdot (-20) = -80 \]

So,

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

Now, we substitute into the quadratic formula:

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

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

Finding the prime factorization of 1104:

\[ 1104 = 16 \cdot 69 = 16 \cdot 3 \cdot 23 = 4^2 \cdot 3 \cdot 23 \] Thus, \(\sqrt{1104} = \sqrt{16 \cdot 69} = 4 \sqrt{69}\)

Now we substitute this back into our formula:

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

Now, simplify it further:

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

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

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

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