# michaelcai

0
84
4
+518

### Let $f(x)=ax^2+bx+a$, where $a$ and $b$ are constants and $a\ne 0$. If one of the roots of the equation $f(x)=0$ is $x=4$, what is the other

michaelcai  Dec 20, 2017
+2
89
6
+518

### Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.

michaelcai  Dec 14, 2017
+2
88
2
+518

### The value $$\left(\frac{1+\sqrt 3}{2\sqrt 2}+\frac{\sqrt 3-1}{2\sqrt 2}i\right)^{72}$$ is a positive real number. What real number is it?

michaelcai  Dec 14, 2017
0
90
5
+518

### Algrebra Help

michaelcai  Dec 11, 2017
+1
62
1
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### Simplify $(i+1)^{3200}-(i-1)^{3200}$

michaelcai  Dec 11, 2017
0
86
2
+518

### Find a complex number $z$ such that the real part and imaginary part of $z$ are both integers, and such that $$z\overline z = 89.$$

michaelcai  Dec 11, 2017
0
80
3
+518

### Compute $1+i+i^2+i^3+i^4+\cdots+i^{2009}$.

michaelcai  Dec 11, 2017
+2
69
1
+518
michaelcai  Dec 4, 2017
+1
90
2
+518
michaelcai  Dec 4, 2017
+1
58
1
+518

### Some functions that aren't invertible can be made invertible by restricting their domains. For example, the function $x^2$ is invertible if

michaelcai  Dec 4, 2017
+2
81
2
+518

### The function $$f(x) = \frac{cx}{2x+3}$$satisfies $f(f(x))=x$ for all real numbers $x\ne -\frac 32$. Find $c$.

michaelcai  Dec 4, 2017
+2
107
6
+518

### Triangle $ABC$ is inscribed in equilateral triangle $PQR$, as shown. If $PC = 3$, $BP = CQ = 2$, and $\angle ACB = 90^\circ$, then compute \$

michaelcai  Dec 4, 2017