Question

$\alpha ,\beta$ are the roots of the equation $x^2$ + bx + c = 0 and $\alpha +2,\beta +2$ are the roots of the equation $x^2$ + px + q=0. If the minimum value of the expression $x^2$ + bx + c is +2, find the minimum value of $x^2$ + px + q

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Answer 1

The graph of $x^2$ + bx + c gets shifted by 2 units when the roots become $\alpha +2and\beta +2$. The minimum value will remain the same.

Below is the graph for $X^2$ - 5x + 6 and $X^2$ - 11x + 30. The roots are 2, 3 and 5, 6 respectively. The roots of the second

equation ($X^2$ - 11x + 30) are obtained by adding 3 to the roots of first equation. You can see in the graph that the

minimum value is same for both the functions.