 Theory of expressions
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# Let $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{2}+{\mathrm{b}}_{1}\mathrm{x}+{\mathrm{c}}_{1},\mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^{2}+{\mathrm{b}}_{2}\mathrm{x}+{\mathrm{c}}_{2}$. Let the real roots of  and real roots of . The least value of f(x) is -1/4. The least value of g(x) occurs at x = 7/2.

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## $\begin{array}{l}\left(\mathrm{\beta }-\mathrm{\alpha }\right)=\left(\left(\mathrm{\beta }+\mathrm{h}\right)-\left(\mathrm{\alpha }+\mathrm{h}\right)\right)\\ \left(\mathrm{\beta }+\mathrm{\alpha }{\right)}^{2}-4\mathrm{\alpha \beta }=\left[\left(\mathrm{\beta }+\mathrm{h}\right)+\left(\mathrm{\alpha }+\mathrm{h}\right){\right]}^{2}-4\left(\mathrm{\beta }+\mathrm{h}\right)\left(\mathrm{\alpha }+\mathrm{h}\right)\\ {\left(-{\mathrm{b}}_{1}\right)}^{2}-4{\mathrm{c}}_{1}={\left(-{\mathrm{b}}_{2}\right)}^{2}-4{\mathrm{c}}_{2}\\ {\mathrm{D}}_{1}={\mathrm{D}}_{2}\end{array}$The least value of f(x) isTherefore, the least value ofThe least value of g(r) occurs atNow, ${\mathrm{b}}_{2}^{2}-4{\mathrm{c}}_{2}={\mathrm{D}}_{2}$or

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