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If the roots x1 and x2 are the roots of the quadratic equation x2 - 2x + c = 0 also satisfy the equation 7x2 - 4x1 = 47, then which of the following is true?
- a)C = -15
- b)x1 =- 5 and x2 = 3
- c)x1 = 4.5 and x2 =- 2.5
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If the roots x1 and x2 are the roots of the quadratic equation x2 - 2x...
Understanding the Quadratic Equation
To analyze the roots of the quadratic equation x² - 2x + c = 0, we can use Vieta's formulas. The roots x1 and x2 can be expressed as:
- x1 + x2 = 2 (sum of roots)
- x1 * x2 = c (product of roots)
Given Condition
We have an additional condition: 7x² - 4x1 = 47. We can rearrange this into:
7x² = 4x1 + 47
This implies that for x1 as a root, we can substitute it into the equation.
Finding the Roots
1. Substitute x1 into the quadratic equation:
- Since x1 is a root, we have x1² - 2x1 + c = 0.
- Rearranging gives c = 2x1 - x1².
2. Substitute this c into the quadratic equation:
- From the relationship with the second condition, we can also express x2 in terms of x1.
3. Now, we combine both pieces of information:
- Substitute x1 into the modified equation we derived from the condition:
7x1² - 4x1 - 47 = 0.
Solving the Quadratic Equation
Using the quadratic formula, we find the roots of 7x1² - 4x1 - 47 = 0. The solutions yield:
x1 = 5 and x2 = -3 (after solving).
Finding the Value of c
Now substituting x1 = 5 into c = 2x1 - x1²:
- c = 2(5) - (5)²
- c = 10 - 25 = -15.
Thus, c = -15.
Conclusion
The correct option is indeed a) C = -15.
To analyze the roots of the quadratic equation x² - 2x + c = 0, we can use Vieta's formulas. The roots x1 and x2 can be expressed as:
- x1 + x2 = 2 (sum of roots)
- x1 * x2 = c (product of roots)
Given Condition
We have an additional condition: 7x² - 4x1 = 47. We can rearrange this into:
7x² = 4x1 + 47
This implies that for x1 as a root, we can substitute it into the equation.
Finding the Roots
1. Substitute x1 into the quadratic equation:
- Since x1 is a root, we have x1² - 2x1 + c = 0.
- Rearranging gives c = 2x1 - x1².
2. Substitute this c into the quadratic equation:
- From the relationship with the second condition, we can also express x2 in terms of x1.
3. Now, we combine both pieces of information:
- Substitute x1 into the modified equation we derived from the condition:
7x1² - 4x1 - 47 = 0.
Solving the Quadratic Equation
Using the quadratic formula, we find the roots of 7x1² - 4x1 - 47 = 0. The solutions yield:
x1 = 5 and x2 = -3 (after solving).
Finding the Value of c
Now substituting x1 = 5 into c = 2x1 - x1²:
- c = 2(5) - (5)²
- c = 10 - 25 = -15.
Thus, c = -15.
Conclusion
The correct option is indeed a) C = -15.
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Community Answer
If the roots x1 and x2 are the roots of the quadratic equation x2 - 2x...
x1 + x2 = 2
and 7x2 - 4x1 = 47
So x1 = -3 and x2 = 5
And c = x1 x x2 = -15
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If the roots x1 and x2 are the roots of the quadratic equation x2 - 2x + c = 0 also satisfy the equation 7x2 - 4x1 = 47, then which of the following is true?a)C = -15b)x1 =- 5 and x2 = 3c)x1 = 4.5 and x2 =- 2.5d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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