Question

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. 3
• D. 4

Answer 1

The Correct Option is C

To find the minimum possible value of the sum of the squares of the roots of the quadratic equation \(x^2 + (a+3)x - (a+5) = 0\), follow these steps:

1. Let the roots of the equation be \( \alpha \) and \( \beta \). Using Vieta's formulas, we have:

\[\alpha+\beta = -(a+3)\]

\[\alpha\beta = -(a+5)\]

2. The sum of the squares of the roots can be expressed as:

\[\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta\]

Substituting Vieta's formulas, we get:

\[ (\alpha+\beta)^2 = (-(a+3))^2 = (a+3)^2 \]

\[ 2\alpha\beta = 2(- (a+5)) = -2(a+5) \]

Therefore,

\[\alpha^2 + \beta^2 = (a+3)^2 + 2(a+5)\]

3. Simplify the expression:

\[(a+3)^2 + 2(a+5) = a^2 + 6a + 9 + 2a + 10 = a^2 + 8a + 19\]

4. To minimize \( a^2 + 8a + 19 \), complete the square:

\[ a^2 + 8a + 19 = (a+4)^2 - 16 + 19 = (a+4)^2 + 3 \]

The minimum value occurs when \( (a+4)^2 = 0 \), i.e., \( a = -4 \).

Substitute \( a = -4 \) into \((a+4)^2 + 3\):

\[ 0 + 3 = 3 \]

Hence, the minimum possible value of the sum of the squares of the roots is \(\mathbf{3}\).