The equations $x^2-a x+b=0$ and $x^2+b x-a=0$ have a common root, then

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Aicharya · Accepted Answer

Equation $x^2-a x+b$ and $x^2+b x-a$ have a common root. Let $\alpha$ be the common root of both quadratic equations $ \Rightarrow \quad \begin{aligned} & \alpha^2-a \alpha+b=0 \ & \alpha^2+b \alpha-a=0 \end{aligned} $ Using cross-multiplication $ \begin{gathered} \frac{\alpha^2}{-a \quad b}=\frac{\alpha}{b \quad 1}=\frac{1}{1-a} \ b-a \quad-a \quad 1 \quad 1 \quad b \ \frac{\alpha^2}{a^2-b^2}=\frac{\alpha}{b+a}=\frac{1}{b+a} \end{gathered} $ We get $ \frac{\alpha}{b+a}=\frac{1}{b+a} \Rightarrow \alpha=1 $ Now, using first two fraction $ \begin{array}{rlrl} & & \frac{1}{a^2-b^2} & =\frac{1}{a+b} \ \Rightarrow & & a+b & =a^2-b^2 \ \Rightarrow & & (a+b) & =(a+b)(a-b) \ \Rightarrow & & (a+b)(a-b-1) & =0 \ & a+b=0 	ext { or } a-b-1 & =0 \Rightarrow a-b=1 \end{array} $

The equations $x^2-a x+b=0$ and $x^2+b x-a=0$ have a common root, then

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