Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

Select the two values of [tex]\( x \)[/tex] that are roots of this equation:

[tex]\[ x^2 + 3x - 3 = 0 \][/tex]

A. [tex]\( x = \frac{-3 + \sqrt{3}}{2} \)[/tex]
B. [tex]\( x = \frac{-3 - \sqrt{3}}{2} \)[/tex]
C. [tex]\( x = \frac{-3 - \sqrt{2i}}{2} \)[/tex]
D. [tex]\( x = \frac{3 + \sqrt{21}}{2} \)[/tex]

Sagot :

To find the values of [tex]\( x \)[/tex] that are the roots of the quadratic equation [tex]\( x^2 + 3x - 3 = 0 \)[/tex], we will follow these steps using the quadratic formula. The quadratic formula is given by:

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

In our equation [tex]\( x^2 + 3x - 3 = 0 \)[/tex], the coefficients are:
[tex]\[ a = 1, \, b = 3, \, c = -3 \][/tex]

1. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-3) = 9 + 12 = 21 \][/tex]

2. Calculate the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a} = \frac{-3 \pm \sqrt{21}}{2} \][/tex]

This yields two roots:
[tex]\[ x_1 = \frac{-3 + \sqrt{21}}{2} \quad \text{and} \quad x_2 = \frac{-3 - \sqrt{21}}{2} \][/tex]

However, from the result given:
[tex]\[ (0.7912878474779199, -3.79128784747792) \][/tex]

we know that these numeric values correspond to approximately:
[tex]\[ 0.791 \quad \text{and} \quad -3.791 \][/tex]

These values need to be compared with the given choices:

- Choice A: [tex]\(\frac{-3+\sqrt{3}}{2} \approx \text{value is not matching}\)[/tex]
- Choice B: [tex]\(\frac{-3-\sqrt{3}}{2} \approx \text{value is not matching}\)[/tex]
- Choice C: [tex]\(\frac{-3-\sqrt{2i}}{2}\)[/tex] involves imaginary number.
- Choice D: [tex]\(\frac{3+\sqrt{21}}{2}\)[/tex].

Given the exact roots were used to ensure there are no errors, we find that the roots are:

Roots Comparison:
- Numerical root [tex]\(\approx 0.79\)[/tex] aligns with [tex]\( \frac{-3+\sqrt{21}}{2} \)[/tex].
- Numerical root [tex]\(\approx -3.79\)[/tex] aligns with [tex]\( \frac{-3-\sqrt{21}}{2} \)[/tex].

Thus, the correct choices are [tex]\((\frac{-3+\sqrt{21}}{2})\)[/tex] and [tex]\((\frac{-3-\sqrt{21}}{2})\)[/tex].

### Conclusion
Given the numeric comparison, the correct choices are neither mentioned in the problem sets. Choices like A, B, C, D should be re-evaluated and corrected as the given numerical values align with roots having denominator difference.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.