Common roots and finding real values

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The discussion centers on finding the real value of \(a\) in the quadratic equations \(ax^2 + 2x + 1 = 0\) and \(x^2 + 2x + a = 0\) that share common roots. By applying Vieta's formulas, it is established that the sum of the roots from both equations leads to the conclusion that \(a = 1\). This solution is derived from equating the expressions for the sum of the roots, resulting in the definitive value of \(a\).

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Sudharaka
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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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