Question

Find the numerical difference of the roots of the equation
​x²​27x21850
​

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{following}\mathrm{equation} \\ {\mathrm{x}}^2-27\mathrm{x}-21850=0 \\ \mathrm{Compare}\mathrm{with}{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0,\mathrm{we}\mathrm{get} \\ \mathrm{a}=1,\mathrm{b}=-27,\mathrm{c}=-21850 \\ \mathrm{Use}\mathrm{quadratic}\mathrm{formula}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}. \\ \mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}} \\ =\frac{-\left(-27\right)\pm \sqrt{{\left(-27\right)}^2-4\left(1\right)\left(-21850\right)}}{2\left(1\right)} \\ =\frac{27\pm \sqrt{729+87400}}{2} \\ =\frac{27\pm \sqrt{88129}}{2} \\ \mathrm{So}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}\mathrm{are}\frac{27}{2}+\frac{\sqrt{88129}}{2},\frac{27}{2}-\frac{\sqrt{88129}}{2} \\ \mathrm{Thus},\mathrm{the}\mathrm{required}\mathrm{numerical}\mathrm{difference}\mathrm{is}, \\ \left(\frac{27}{2}+\frac{\sqrt{88129}}{2}\right)-\left(\frac{27}{2}-\frac{\sqrt{88129}}{2}\right)=\sqrt{88129} \end{gathered}$$

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