Question

Solve the quadratic equation.
x2+4x+2=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 \(x^2 + 4x + 2 = 0\), we can use the quadratic formula, which is given by:

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

In the equation \(ax^2 + bx + c = 0\), we have:

- \(a = 1\)
- \(b = 4\)
- \(c = 2\)

Now we can plug these values into the quadratic formula.

First, we compute the discriminant (\(b^2 - 4ac\)):

\[
b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 2 = 16 - 8 = 8
\]

Since the discriminant is positive (\(8 > 0\)), there are two distinct real solutions.

Now substituting \(a\), \(b\), and the discriminant into the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{8}}{2 \cdot 1}
\]

Since \(\sqrt{8} = 2\sqrt{2}\), we can simplify further:

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

This simplifies to:

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

Thus, the two solutions are:

1. \(x_1 = -2 + \sqrt{2}\)
2. \(x_2 = -2 - \sqrt{2}\)

You can enter these values in the boxes provided. Here are the solutions:

- Exact solution 1: \(-2 + \sqrt{2}\)
- Exact solution 2: \(-2 - \sqrt{2}\)