Question

Select the two values of x that are roots of this equation.
x2+2x-4=0
0x=-1+V5
0x=-1- 2V5
0x=-1-V5
0x=-1+2V5

Answer 1

To find the roots of the equation \( x^2 + 2x - 4 = 0 \), we can use the quadratic formula:

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

In the equation \( x^2 + 2x - 4 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 2 \)
- \( c = -4 \)

Now, let's calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = 2^2 - 4(1)(-4) = 4 + 16 = 20
\]

Next, we can find the roots using the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{20}}{2(1)} = \frac{-2 \pm \sqrt{20}}{2}
\]

We can simplify \( \sqrt{20} \):

\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
\]

Substituting this back into the equation for \( x \):

\[
x = \frac{-2 \pm 2\sqrt{5}}{2} = -1 \pm \sqrt{5}
\]

This gives us the two roots:

\[
x_1 = -1 + \sqrt{5}, \quad x_2 = -1 - \sqrt{5}
\]

Now comparing with the provided options, the two roots we get can be represented as:

1. \( 0x = -1 + V5 \) (where \( V5 \) is shorthand for \( \sqrt{5} \))
2. \( 0x = -1 - V5 \)

Thus, the two values of \( x \) that are roots of the equation are:

- \( x = -1 + \sqrt{5} \)
- \( x = -1 - \sqrt{5} \)