Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
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.
[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're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
