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α , β are the...

Question

$α, β$ are the roots of the equation $x^{2} + b x + c = 0$ and $α + 2, β + 2$ are the roots of the equation $x^{2} + p x + q = 0$ . If the minimum value of the expression $x^{2} + b x + c$ is $2$ , find the minimum value of $x^{2} + p x + q$

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Solution

The graph of $x^{2}$ + bx + c gets shifted by 2 units when the roots become $α + 2$ and $β + 2$ . The minimum value will remain the same.

For example:
The graph for $x^{2} - 5 x + 6$ and $x^{2} - 11 x + 30$ . The roots are $2, 3$ and $5, 6$ respectively.
The roots of the second equation ( $x^{2} - 11 x + 30)$ 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.

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α,β are the roots of the equation x2+bx+c=0 and α+2,β+2 are the roots of the equation x2+px+q=0. If the minimum value of the expression x2+bx+c is 2, find the minimum value of x2+px+q

$α, β$ are the roots of the equation $x^{2} + b x + c = 0$ and $α + 2, β + 2$ are the roots of the equation $x^{2} + p x + q = 0$ . If the minimum value of the expression $x^{2} + b x + c$ is $2$ , find the minimum value of $x^{2} + p x + q$