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If the roots x1 and x2 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, x2 = 3
- c)x1 = 4.5, 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 of the quadratic equation x2- 2x + c = 0 also ...
7x2 – 4x1 = 47
x1 + x2 = 2
Solving 11x2 = 55
x2 = 5 & x1 = - 3
∴ c = - 15
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Community Answer
If the roots x1 and x2 of the quadratic equation x2- 2x + c = 0 also ...
To solve this problem, let's first find the values of x1 and x2 using the quadratic equation x^2 - 2x + c = 0.
We know that the sum of the roots of a quadratic equation is given by the formula -b/a and the product of the roots is given by the formula c/a, where a, b, and c are the coefficients of the quadratic equation.
In this case, the sum of the roots is 2 and the product of the roots is c. Therefore, we have the following equations:
x1 + x2 = 2
x1 * x2 = c
Next, we are given that the roots also satisfy the equation 7x^2 - 4x1 = 47. Substituting the values of x1 and x2 into this equation, we get:
7(x1^2 + x2^2) - 4(x1 + x2) = 47
Simplifying this equation, we have:
7(x1^2 + x2^2) - 4(2) = 47
7(x1^2 + x2^2) - 8 = 47
7(x1^2 + x2^2) = 55
x1^2 + x2^2 = 55/7
Now, let's consider the options given:
a) c = -15
If c = -15, then x1 * x2 = -15. But x1 * x2 = c, so this option is not valid.
b) x1 = -5, x2 = 3
If x1 = -5 and x2 = 3, then x1 + x2 = -2, which is not equal to 2. Therefore, this option is not valid.
c) x1 = 4.5, x2 = -2.5
If x1 = 4.5 and x2 = -2.5, then x1 + x2 = 2, which is equal to 2. Also, x1^2 + x2^2 = 4.5^2 + (-2.5)^2 = 20.5, which is not equal to 55/7. Therefore, this option is not valid.
d) None of these
We have already found that option b and c are not valid. Therefore, the correct answer is option a.
In conclusion, the only valid option is a) c = -15.
We know that the sum of the roots of a quadratic equation is given by the formula -b/a and the product of the roots is given by the formula c/a, where a, b, and c are the coefficients of the quadratic equation.
In this case, the sum of the roots is 2 and the product of the roots is c. Therefore, we have the following equations:
x1 + x2 = 2
x1 * x2 = c
Next, we are given that the roots also satisfy the equation 7x^2 - 4x1 = 47. Substituting the values of x1 and x2 into this equation, we get:
7(x1^2 + x2^2) - 4(x1 + x2) = 47
Simplifying this equation, we have:
7(x1^2 + x2^2) - 4(2) = 47
7(x1^2 + x2^2) - 8 = 47
7(x1^2 + x2^2) = 55
x1^2 + x2^2 = 55/7
Now, let's consider the options given:
a) c = -15
If c = -15, then x1 * x2 = -15. But x1 * x2 = c, so this option is not valid.
b) x1 = -5, x2 = 3
If x1 = -5 and x2 = 3, then x1 + x2 = -2, which is not equal to 2. Therefore, this option is not valid.
c) x1 = 4.5, x2 = -2.5
If x1 = 4.5 and x2 = -2.5, then x1 + x2 = 2, which is equal to 2. Also, x1^2 + x2^2 = 4.5^2 + (-2.5)^2 = 20.5, which is not equal to 55/7. Therefore, this option is not valid.
d) None of these
We have already found that option b and c are not valid. Therefore, the correct answer is option a.
In conclusion, the only valid option is a) c = -15.
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If the roots x1 and x2 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, x2 = 3c)x1 = 4.5, x2 = -2.5d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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