Question

Find the solutions of the following quadratic equation. x^2-4x-6=0 (1 point) x=4+sqrt(10) or x=4-sqrt(10) x=2+sqrt(10) or x=2-sqrt(10) x=2+sqrt(6) or x=2-sqrt(6) x=4+sqrt(6) or x=4-sqrt(6)

Answer 1

To solve the quadratic equation \( x^2 - 4x - 6 = 0 \), we can use the quadratic formula:

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

where \( a = 1 \), \( b = -4 \), and \( c = -6 \).

First, we calculate \( b^2 - 4ac \):

\[
b^2 = (-4)^2 = 16
\]
\[
4ac = 4 \cdot 1 \cdot (-6) = -24
\]
\[
b^2 - 4ac = 16 - (-24) = 16 + 24 = 40
\]

Now we can substitute \( a \), \( b \), and \( b^2 - 4ac \) into the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{40}}{2 \cdot 1} = \frac{4 \pm \sqrt{40}}{2}
\]

Next, we simplify \( \sqrt{40} \):

\[
\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}
\]

So we substitute this back into the formula:

\[
x = \frac{4 \pm 2\sqrt{10}}{2}
\]

This simplifies to:

\[
x = 2 \pm \sqrt{10}
\]

Thus, the solutions are:

\[
x = 2 + \sqrt{10} \quad \text{or} \quad x = 2 - \sqrt{10}
\]

The correct answer is:

**x=2+sqrt(10) or x=2-sqrt(10)**