Question

Select the two values of x that are roots of this equation.
х2+3x-6=0
•x=
-3+ V15
2
•x=
=3-v33
0 x ==3-VI5
0х =
-3+V33
2

Answer 1

To find the roots of the equation \( x^2 + 3x - 6 = 0 \), we can use the quadratic formula:

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

In this case, \( a = 1 \), \( b = 3 \), and \( c = -6 \). Plugging these values into the formula:

1. Calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 - 4ac = 3^2 - 4(1)(-6) = 9 + 24 = 33
\]

2. Now substitute back into the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{33}}{2}
\]

From this, we can identify the two potential roots as:
- \( x = \frac{-3 + \sqrt{33}}{2} \)
- \( x = \frac{-3 - \sqrt{33}}{2} \)

Now comparing these results to the options provided:
1. \( x = -3 + \frac{\sqrt{15}}{2} \)
2. \( x = 3 - \sqrt{33} \)
3. \( x = 3 - \frac{\sqrt{15}}{2} \)
4. \( x = -3 + \frac{\sqrt{33}}{2} \)

The correct roots are:
- \( x = \frac{-3 + \sqrt{33}}{2} \)
- \( x = \frac{-3 - \sqrt{33}}{2} \)

From the list, the corresponding options that match this format logically lead to:
- \( x = -3 + \frac{\sqrt{33}}{2} \)
- \( x = -3 - \frac{\sqrt{33}}{2} \)

The appropriate selections are:
- \( x = \frac{-3 + \sqrt{33}}{2} \)
- \( x = \frac{-3 - \sqrt{33}}{2} \)

Thus, the two roots are:
- \( x = -3 + \frac{\sqrt{33}}{2} \)
- \( x = -3 - \frac{\sqrt{33}}{2} \)