Question

Solve the quadratic equation.
x2+12x+1=0
Write one exact solution in each box. You can add or remove boxes. If there are no real solutions, remove all boxes.

Answer 1

To solve the quadratic equation \( x^2 + 12x + 1 = 0 \), we can use the quadratic formula:

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

Here, \( a = 1 \), \( b = 12 \), and \( c = 1 \).

Now, we can calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = 12^2 - 4 \times 1 \times 1 = 144 - 4 = 140
\]

Since the discriminant is positive (140 > 0), there are two distinct real solutions.

Using the quadratic formula:

\[
x = \frac{-12 \pm \sqrt{140}}{2 \cdot 1}
\]

Now we simplify \( \sqrt{140} \):

\[
\sqrt{140} = \sqrt{4 \times 35} = 2\sqrt{35}
\]

Now substituting this back into the formula:

\[
x = \frac{-12 \pm 2\sqrt{35}}{2}
\]

This simplifies to:

\[
x = -6 \pm \sqrt{35}
\]

Hence, the two solutions are:

\[
x_1 = -6 + \sqrt{35} \quad \text{and} \quad x_2 = -6 - \sqrt{35}
\]

You can write the exact solutions in the boxes as follows:

1. First solution: \( -6 + \sqrt{35} \)
2. Second solution: \( -6 - \sqrt{35} \)