If p and q`(!=0)`are the roots of the equation`x^2+px+q=0`, then the least value of `x^2+px+q(x inR)` is :

Options

A. `-1/4`
B. `1/4`
C. `(-9)/4`
D. 8

Answer

To find the least value of the expression \( x^2 + px + q \) given that \( p \) and \( q \) are the roots of the equation \( x^2 + px + q = 0 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \( x^2 + px + q \). To find its minimum value, we can complete the square.

### Step 2: Completing the square
The expression can be rewritten as:
\[ x^2 + px + q = \left(x + \frac{p}{2}\right)^2 - \frac{p^2}{4} + q \]
This transformation helps us identify the minimum value of the quadratic.

### Step 3: Identify the minimum value
The minimum value of \( \left(x + \frac{p}{2}\right)^2 \) is \( 0 \) (since the square of any real number is non-negative). Thus, the minimum value of the entire expression occurs when:
\[ \left(x + \frac{p}{2}\right)^2 = 0 \]
This gives us:
\[ -\frac{p^2}{4} + q \]

### Step 4: Use the properties of roots
Since \( p \) and \( q \) are the roots of the equation \( x^2 + px + q = 0 \), we can use Vieta's formulas:
- The sum of the roots \( p + q = -p \)
- The product of the roots \( pq = q \)

From the sum of the roots, we can express \( q \) in terms of \( p \):
\[ q = -p - p = -2p \]

### Step 5: Substitute \( q \) back into the expression
Substituting \( q \) into the expression for the minimum value:
\[ -\frac{p^2}{4} + (-2p) = -\frac{p^2}{4} - 2p \]

### Step 6: Find the critical points
To find the minimum of the expression \( -\frac{p^2}{4} - 2p \), we can differentiate it with respect to \( p \) and set the derivative to zero:
\[ \frac{d}{dp}\left(-\frac{p^2}{4} - 2p\right) = -\frac{p}{2} - 2 = 0 \]
Solving for \( p \):
\[ -\frac{p}{2} = 2 \implies p = -4 \]

### Step 7: Calculate the minimum value
Now substituting \( p = -4 \) back into the expression for \( q \):
\[ q = -2(-4) = 8 \]
Now substituting \( p \) and \( q \) back into the minimum value expression:
\[ -\frac{(-4)^2}{4} - 2(-4) = -\frac{16}{4} + 8 = -4 + 8 = 4 \]

### Step 8: Conclusion
The least value of \( x^2 + px + q \) is \( 4 \).