Let $a$ and $b$ be the two roots of $x^2 - 3x -1=0$. Try to solve the following problems without computing $a$ and $b$:
1) Find a quadratic equation whose roots are $a^2$ and $b^2$
2) Find the value of $\frac{1}{a+1}+\frac{1}{b+1}$
3) Find the recursion relationship of $x_n=a^n + b^n$
Find as many different solutions as possible.
Three of the roots of $x^4 + ax^2 + bx + c = 0$ are $2$, $−3$, and $5$. Find the value of $a + b + c$.
In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that
$$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$
Suppose $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are all positive real numbers. If both $a_1x^2 +b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=$ are solvable in real numbers. Show that their roots must be all negative. Furthermore, prove equation $a_1a_2x^2+b_1b_2x+c_1c_2=0$ has two negative real roots too.
Let $x$, $y$, and $z$ be real numbers satisfying $x=6-y$ and $z^2=xy-9$. Show that $x=y$.
Let $\alpha_n$ and $\beta_n$ be two roots of equation $x^2+(2n+1)x+n^2=0$ where $n$ is a positive integer. Evaluate the following expression $$\frac{1}{(\alpha_3+1)(\beta_3+1)}+\frac{1}{(\alpha_4+1)(\beta_4+1)}+\cdots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}$$
Let real numbers $a$, $b$, and $c$ satisfy
$$
\left\{
\begin{array}{rcl}
a^2 - bc-8a +7&=&0\\
b^2 + c^2 +bc-6a+6&=&0
\end{array}
\right.
$$
Show that $1 \le a \le 9$.
Find one real solution $(a, b, c, d)$ to the following system:
$$
\left\{
\begin{array}{rcl}
a+b+c+d&=&-2\\
ab+ac+ad+bc+bd+cd&=&-3\\
abc+abd+acd+bcd&=&4\\
abcd&=&3
\end{array}
\right.
$$
If $m^2 = m+1, n^2-n=1$ and $m\ne n$, compute $m^7 +n^7$.
Find the range of real number $a$ if the two roots of $x^2+2ax+6-a=0$ satisfy one of the following condition:
- two roots are both greater than 1
- one root is greater than 1 and the other is less than 1
Solve equation $(6x+7)^2(3x+4)(x+1)=6$ in real numbers.
If $x^2 + 11x+16=0, y^2 + 11y+16=0$, and $x\ne y$, what is the value of $$\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}$$
Let $x_1$ and $x_2$ be two real roots of $x^2-x-1=0$. Find the value of $2x_1^5 + 5x_2^3$.
Find integer $m$ such that the equation $x^2+mx-m+1=0$ has two positive integer roots.
Let $\alpha$ and $\beta$ be two real roots of $x^4 +k=3x^2$ and also satisfy $\alpha + \beta = 2$. Find the value of $k$.
Determine all roots, real or complex, of the following system
\begin{align}
x+y+z &= 3\\
x^2+y^2+z^2 &= 3\\
x^3+y^3+z^3 &= 3
\end{align}
Given that $x^2+5x+6=20$, find the value of $3x^2 + 15x+17$.
Let $r_1, \cdots, r_5$ be the roots of the polynomial $x^5 + 5x^4 - 79x^3 +64x^2 + 60x+144$. What is $r_1^2 +\cdots + r_5^2$?
Find all pairs of real numbers $(a, b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Suppose that $P(x)$ is a polynomial with the property such that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is quintic (i.e. has a degree of $5$) with roots $r_1, \cdots, r_5$, find all the possible values of $|r_1|+|r_2|+\cdots+|r_5|$.
Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.
The curve $y=x^4+2x^3-11x^2-13x+35$ has a bitangent (a line tangent to the curve at two points). What is the equation of this bitangent line.
Solve this equation $(x-2)(x+1)(x+4)(x+7)=19$.
Let real numbers $x, y,$ and $z$ satisfy $$x+\frac{1}{y}=4\quad\text{,}\quad y+\frac{1}{z}=1\quad\text{,}\quad z +\frac{1}{x}=\frac{7}{3}$$ Find the value of $xyz$.
Find the range of real number $a$ if equation $\mid\frac{x^2}{x-1}\mid=a$ has exactly two distinct real roots.