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Assume that $Δ=b^2−4ac≈0$ means that the actual value is $Δ=k^2·\mu$ where $μ=2^{-52}\sim 10^{-15}$ is the machine constant for the double type. Then the floating point error for the computation of $fl(Δ)=fl(fl(b⋅b)−4⋅fl(a⋅c))$ and assuming "normal" values for $a,b,c$, for instance $a,b,c≈1,\pm2,1$, is of size $fl(Δ)-Δ=m·μ$ where $|m|<2$. Since all ...


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I suppose that you solve for the roots of $a+bx+cx^2=0$ in which $\Delta=b^2-4ac$ is small. The roots are given by $$x_{\pm}=\dfrac {-b \pm \sqrt {b^2 - 4ac}} {2c}$$ Let us consider the case where $b>0$; so, by absolute value, the largest root is $$x_{-}=-\dfrac {b + \sqrt {b^2 - 4ac}} {2c}$$ and this should not make problem. Now, remember that ...


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Don't solve the equation. Instead, plot a picture of $f(x):=xe^x$ for $x\in\mathbb R$. Where is $f(x)$ positive, negative? Use calculus to find where $f(x)$ is increasing, decreasing. What is the behavior of $f(x)$ as $x\to\infty$, or $x\to-\infty$? Once you've plotted $y=f(x)$, intersect it with the horizontal line $y=a$. Note that $a=-1/e$ is related to a ...



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