MaxWong

What is the definition of \(\oplus\) here?

Let a, b be real numbers and suppose \((a + bi)^2 = 21-20i\\\).

\( a^2 - b^2 + 2abi = 21-20i\\ \begin{cases}a^2 - b^2 = 21\\2ab = -20\end{cases}\)

Solving this system, \(\begin{cases}a = 5\\b = -2\end{cases}\) or \(\begin{cases}a = -5\\b = 2\end{cases}\).

The square roots are \(-5+2i\) and \(5-2i\).

You can change both of the recurring decimals to fractions.

In fact,

\(0.\overline{72} = \dfrac{8}{11}\) and \(0.\overline{27} = \dfrac{3}{11}\)

Therefore the answer is \(\dfrac83\).

Note that \(12 \times 90 = 1080\).

Then, \(1089 = 12\times 90 + 9\).

Taking mod 12 on both sides results in \(1089 \equiv 9 \pmod{12}\).

https://web2.0calc.com/questions/pls-help-due-tmrow_14#r3

This one is a very similar problem, solved by CPhill. Hope this helps.

Do you mean \(\dfrac{x^2}{2x - 1}\) and \(\dfrac{x + 1}{x + 9}\)?

Add the two equations and you will get \(\dfrac2x = -4\). Then you can solve for x.

Similarly you can solve for y. Then just add them together.

We use a geometric approach because we see that if a and b are side lengths of a triangle and the included angle is 60 degrees, then the remaining side is \(\sqrt{a^2 - ab + b^2}\). Also, if the included angle is 120 degrees instead, then the remaining side is \(\sqrt{a^2 + ab + b^2}\).

(Please try to prove this using Law of Cosines.)

Refer to the following diagram:

Using triangle inequality on \(\triangle BCD\) yields the required inequalities immediately.

Exercise: Try to figure out the case when equality holds.

By factor theorem, if a polynomial f(x) leaves a remainder of r when divided by x - a, then f(a) is the remainder.

Therefore, we have

\(\begin{cases}1^4 + a\cdot 1^3 + b\cdot 1 + c = 14\\ (-1)^4 + a\cdot (-1)^3 + b\cdot (-1) + c = 0\\ (-2)^4 + a\cdot (-2)^3 + b\cdot (-2) + c = -28\end{cases}\)

Simplifying,

\(\begin{cases}1+ a+b+c= 14\\ 1-a-b+c = 0\\ 16-8a-2b+ c = -28\end{cases}\)

If you solve the system, you will get \((a,b,c) = (6,1,6)\). Please try to fill in the solving process on your own.

P.S. You can solve the system by method of elimination, method of substitution, Gaussian elimination, or whatever method you know.

\(\begin{cases}725x + 727y = 1500 \text{ --- }(1)\\729x + 7y = 1508\text{ --- }(2)\end{cases}\)

Subtract (1) from (2) to get 

\(4x - 720y = 8\\ x - 180y = 2\text{ --- }(3)\)

Now, solve the system formed by equations (2) and (3) using method of elimination, and you will get \((x, y) = \left(\dfrac{271454}{131227}, \dfrac{50}{131227}\right)\)

Therefore, x - y = \(\dfrac{271404}{131227}\), as robloxDying suggested.