Find all real solutions for the system [MATH]4x^2-40\left\lfloor{x}\right\rfloor+51=0[/MATH] where [MATH]\left\lfloor{x}\right\rfloor[/MATH] represents the floor of $x$.
My solution:
First, notice that if we rewrite the equality as [MATH]4x^2+51=40\left\lfloor{x}\right\rfloor[/MATH], we can tell [MATH]\left\lfloor{x}\right\rfloor[/MATH] must be a positive figure.
Find the number of real solutions for the system $x^4-x^3+x^2-4x-12=0$.
Find the number of real solutions for the system $x^4-x^3+x^2-4x-12=0$.
First, we let $f(x)=x^4-x^3+x^2-4x-12$. $f(x)$ clearly is a quartic function and it has at most 4 real roots.
If we can factorize $f(x)$ as $f(x)=(x-a)(x-b)(x-c)(x-d)$, then $f(x)$ has 4 real roots.
First, we let $f(x)=x^4-x^3+x^2-4x-12$. $f(x)$ clearly is a quartic function and it has at most 4 real roots.
If we can factorize $f(x)$ as $f(x)=(x-a)(x-b)(x-c)(x-d)$, then $f(x)$ has 4 real roots.
Analysis Quiz 19: Multiple-Choice Test (Improve Analytical Skill)
Question 1: If you're asked to simplify [MATH]\left(1+\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}\right)^2[/MATH], do you think by turning the $1$ as [MATH]\left(\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}\right)[/MATH] is feasible in order to simplify the expression?
A. Yes.
B. No.
A. Yes.
B. No.
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