Question

Find the solutions of the following quadratic equation
X^2 -4x -6=0

Answer 1

To find the solutions of the quadratic equation \( x^2 - 4x - 6 = 0 \), you can use the quadratic formula:

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

In this equation, the coefficients are:
- \( a = 1 \)
- \( b = -4 \)
- \( c = -6 \)

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

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

Now insert the values into the quadratic formula:

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

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

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

Now, substitute this back into the equation:

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

Simplify further:

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

Thus, the solutions to the equation \( x^2 - 4x - 6 = 0 \) are:

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