Answer :
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.
[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.
