Practice (13)
Solve this equation $2x^4 + 3x^3 -16x^2+3x + 2 =0$.
Solve this equation: $(x^2-x-1)^{x+2}=1$.
Let real numbers $a, b, c, d$ satisfy
$$
\left\{
\begin{array}{ccl}
ax+by&=3\\
ax^2+by^2&=7\\
ax^3+by^3&=16\\
ax^4 + by^4 &=42
\end{array}
\right.
$$
Find $ax^5+by^5$.
If all roots of the equation $$x^4-16x^3+(81-2a)x^2 +(16a-142)x+(a^2-21a+68)=0$$ are integers, find the value of $a$ and solve this equation.
How many pairs of ordered real numbers $(x, y)$ are there such that
$$
\left\{
\begin{array}{ccl}
\mid x\mid + y &=& 12\\
x + \mid y \mid &=&6
\end{array}
\right.
$$
Let $a$, $b$, and $c$ be three distinct numbers such that $$\frac{a+b}{a-b}=\frac{b+c}{2(b-c)}=\frac{c+a}{3(c-a)}$$
Prove that $8a + 9b + 5c = 0$.
Solve this equation in real numbers: $$\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{1}{2}\times(x+y+z)$$
Let $a$, $b$, and $c$ be the lengths of $\triangle{ABC}$'s three sides. Compute the area of $\triangle{ABC}$ if the following relations hold: $$\frac{2a^2}{1+a^2}=b,\qquad \frac{2b^2}{1+b^2}=c,\qquad \frac{2c^2}{1+c^2}=a$$
Solve this equation $$2\sqrt{2}x^2 + x -\sqrt{1-x ^2}-\sqrt{2}=0$$
Solve the equation
$$x^4-97x^3+2012x^2-97x+1=0$$
Solve the equation $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
For each ordered pair of real numbers $(x,y)$ satisfying\[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that\[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.
There are integers $a$, $b$, and $c$, each greater than 1, such that\[\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]for all $N > 1$. What is $b$?
Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} + \sqrt{x} - 2x - 1=0$$
Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} + x - 1=0$$
Find all the real values of $x$ that satistify: $$\sqrt{3x^2 + 1} - 2\sqrt{x} - x + 1=0$$
Solve $x^{x^{88}} - 88=0$.