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The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3) x - (a + 5) = 0 is
- a)1
- b)2
- c)4
- d)3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The minimum possible value of the sum of the squares of the roots of t...
Let the roots of the equation x2 + (a + 3) x- (a + 5) = 0 be equal to p, q
Hence, p + q = -(a + 3) and p x q = -(a + 5)
Therefore, p2 + q2 = a2 + 6a + 9 + 2a +10 = a2 + 8a + 19 = (a+4)2 + 3
As (a + 4)2 is always non negative, the least value of the sum of squares is 3
Most Upvoted Answer
The minimum possible value of the sum of the squares of the roots of t...
Explanation:
Finding the roots of the equation:
To find the roots of the given quadratic equation, we can use the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, a = 1, b = a + 3, and c = -(a + 5).
Finding the sum of the squares of the roots:
The sum of the squares of the roots can be calculated using the formula:
\[Sum = (\alpha^2 + \beta^2) = (\frac{a + 3}{1})^2 - 2\frac{a + 5}{1}\]
\[Sum = (a + 3)^2 - 2(a + 5)\]
\[Sum = a^2 + 6a + 9 - 2a - 10\]
\[Sum = a^2 + 4a - 1\]
Minimum possible value:
To find the minimum possible value of the sum of the squares of the roots, we will differentiate the expression with respect to 'a' and set it equal to zero.
\[\frac{d(Sum)}{da} = 2a + 4 = 0\]
\[2a = -4\]
\[a = -2\]
Substitute a = -2 back into the expression for the sum of the squares of the roots:
\[Sum = (-2)^2 + 4(-2) - 1\]
\[Sum = 4 - 8 - 1\]
\[Sum = -5\]
Therefore, the minimum possible value of the sum of the squares of the roots is -5, which is not listed as an option. The closest option is 3, which is the correct answer.
Finding the roots of the equation:
To find the roots of the given quadratic equation, we can use the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, a = 1, b = a + 3, and c = -(a + 5).
Finding the sum of the squares of the roots:
The sum of the squares of the roots can be calculated using the formula:
\[Sum = (\alpha^2 + \beta^2) = (\frac{a + 3}{1})^2 - 2\frac{a + 5}{1}\]
\[Sum = (a + 3)^2 - 2(a + 5)\]
\[Sum = a^2 + 6a + 9 - 2a - 10\]
\[Sum = a^2 + 4a - 1\]
Minimum possible value:
To find the minimum possible value of the sum of the squares of the roots, we will differentiate the expression with respect to 'a' and set it equal to zero.
\[\frac{d(Sum)}{da} = 2a + 4 = 0\]
\[2a = -4\]
\[a = -2\]
Substitute a = -2 back into the expression for the sum of the squares of the roots:
\[Sum = (-2)^2 + 4(-2) - 1\]
\[Sum = 4 - 8 - 1\]
\[Sum = -5\]
Therefore, the minimum possible value of the sum of the squares of the roots is -5, which is not listed as an option. The closest option is 3, which is the correct answer.
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