MHB Common roots and finding real values

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The discussion centers on finding the real value of \(a\) for the quadratic equations \(ax^2+2x+1=0\) and \(x^2+2x+a=0\) that share common roots. By applying Vieta's formulas, the sum of the roots from both equations leads to the equation \(-\frac{2}{a}=-2\). Solving this results in \(a=1\). The response notes that a more detailed explanation of the derivation would have been beneficial. The conclusion confirms that the correct value of \(a\) is indeed 1.
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srirahulan's question titled "Algeb" from Math Help Forum,

If \(ax^2+2x+1=0\mbox{ and }x^2+2x+a=0\) have the common roots, find the real value of a.

Hi srirahulan,

Let \(\alpha\mbox{ and }\beta\) be the two roots of these quadratic equations. Then, according to the first equation,

\[\alpha+\beta=-\frac{2}{a}~~~~~~(1)\]

Considering the second equation,

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

By (1) and (2);

\[-\frac{2}{a}=-2\]

\[\therefore a=1\]
 
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