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If $\alpha, \beta$ are the roots of the equation $x^2+b x+c=0$ and $\alpha+h, \beta+h$ are the roots of the equation $x^2+q x+r=0$, then $h$ is equal to
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Verified Answer
The correct answer is:
$\frac{1}{2}(b-q)$
Given that $\alpha$ and $\beta$ are the roots of the equation
$\begin{array}{r}x^2+b x+c=0 \\ \alpha+\beta=-b \text { and } \alpha \beta=c\end{array}$
Also $\alpha+h$ and $\beta+h$ are the roots of the equations
We have,
$x^2+q x+r=0$
$\begin{array}{rlrl} & \therefore & \alpha+h+\beta+h & =-q \\ -b+2 h & =-q \\ \Rightarrow & h & =\frac{(b-q)}{2}\end{array}$
$\begin{array}{r}x^2+b x+c=0 \\ \alpha+\beta=-b \text { and } \alpha \beta=c\end{array}$
Also $\alpha+h$ and $\beta+h$ are the roots of the equations
We have,
$x^2+q x+r=0$
$\begin{array}{rlrl} & \therefore & \alpha+h+\beta+h & =-q \\ -b+2 h & =-q \\ \Rightarrow & h & =\frac{(b-q)}{2}\end{array}$
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