All Answers 358184 Answers

If a is a root, then

a^2 + a(a) + b = 0

2a^2 + b = 0

b = -2a^2    (1)

If b is a root, then

b^2 + ab + b = 0    (2)

Sub (1) into (2)

(-2a^2)^2 +a(-2a^2) + (-2a^2) = 0    simplify

4a^4 -2a^3 - 2a^2 = 0   factor

(2a^2) (2a^2 - a - 1) = 0

Set both factors to 0 and solve

2a^2 = 0        (2a + 1) (a - 1) = 0

a = 0                a = -1/2       a = 1

Using (1).....

When a = 0, b = 0     (trivial solution)

When a = -1/2 , b = -1/2

When  a = 1, b = -2

Proof

x^2 + 0x +0 = 0  .......has the root x = 0.....(a,b) = (0,0)

x^2 - (1/2)x - 1/2 = 0     multiply through by 2

2x^2  - x^2 - 1  =  0  factor

(2x + 1) (x - 1) = 0

Has the roots  x = -1/2  and x  = 1 

So  .....(a,b) = (-1/2, -1/2)

x^2 -2x + 1 = 0      factor

(x -1)  (x -1) = 0.......so x = 1 is a root......(a, b) = (1, -2)

So......

(a, b)  = (0, 0)    (trivial)

(a, b) = (-1/2, -1/2)

(a, b) = (1, -2)

Enter 125^(1/3) to get OneNote to calculate the cube root.

(x+6)(x+7) = factored form

Please rewrite your equation as a expression. 

You know there's a calculator on this site right? 

I think you are missing part of the equation. 

Sorry, I'm not sure.

You could search up how to make the cube root command.

You are very welcome!

:P

I'm not sure what that stuff is, but I think you are wanting \(21\times 24343535\) which equals 511214235.

You are very welcome!

:P

Question:  how do i get to sin^-1

If you mean on the web2.0 calc then ...

Press 2nd and then press asin

asin stands for arc sine which is the same as inverse sine.

Make sure you have it set to degrees or to radians.  There is a toggle button on the bottom of the calc. :)

its 17689

\(133*133=\left \{ 17689 \right \}\)

2)
Given that m and n are positive integers such that m=6 (mod 9) and n=0 (mod 9),
what is the largest integer that mn is necessarily divisible by?

\(\begin{array}{|rcll|} \hline m & \equiv & 6\pmod{9} \\ \text{or} \\ m &=& 6 + z\cdot 9 ,\quad z \in \mathbb{Z} \\\\ n & \equiv & 0\pmod{9} \\ \text{or} \\ n &=& 0 + z_2 \cdot 9, \quad z_2 \in \mathbb{Z} \\\\ \hline \\ mn &=& (6 + z\cdot 9)\cdot z_2 \cdot 9 \\ mn &=& 3\cdot(2 + z\cdot 3)\cdot z_2 \cdot 9 \\ \mathbf{mn} &\mathbf{=}& \mathbf{{\color{red}27}\cdot(2 + 3z)\cdot z_2} \\ \hline \end{array}\)

The largest integer is 27

1) 

We have positive integers a, b and c such that a>b>c.

When a, b and c are divided by 19, the remainders are 4, 2 and 18 respectively.

When the number 2a+b-c is divided by 19, what is the remainder?

\(\begin{array}{|rcll|} \hline a & \equiv & 4\pmod{19} \quad & | \quad \cdot 2 \\ \mathbf{2a} & \mathbf{\equiv} & \mathbf{8\pmod{19}} \\\\ \mathbf{b} & \mathbf{\equiv} & \mathbf{2\pmod{19}} \\\\ c & \equiv & 18\pmod{19} \\ c & \equiv & 18-19\pmod{19} \\ \mathbf{c} & \mathbf{\equiv} & \mathbf{-1\pmod{19} } \\ \\ \hline \\ 2a+b-c & \equiv & 8+2-(-1) \pmod{19} \\ \mathbf{2a+b-c} & \mathbf{\equiv} & \mathbf{{\color{red}11} \pmod{19}} \\ \hline \end{array}\)

The remainder is 11

Here's another way of looking at 1).

a) a=0 makes f(x) a constant function

b)  \(\text{the "top" of } f(x) \text{ is }f(0)=0 \\\text{ the "top" of }g(x) \text{ is } f\left(\dfrac b 6\right) =-\dfrac{b^2}{12} \)

c)  \(a(-3)^2 + 8 = 5\\ 9a = -3\\ a=-\dfrac 1 3\)

d)  \(\text{the coefficient of the }x^2 \text{ term controls the narrowness of the parabola}\\ \text{the larger absolute value of the coefficient the narrower the parabola}\\ \text{a negative value produces a downward facing parabola, thus}\\ a < -3 \text{ will produce a downward facing parabola that is narrower than g(x)}\)

e) \(3(-2)^2 - b(-2) = 0\\ 12 +2 b = 0\\ b=-6\)

deleted.

a hitachi magic wand?

2, 11, 20, 29...
Which term in this sequence is 227?

\(\begin{array}{|rcll|} \hline & \text{difference} \\ \hline a_1=2 \\ & \color{red}9 \\ a_2=11 \\ & \color{red}9 \\ a_3=20 \\ & \color{red}9 \\ a_4=29 \\ \ldots \\ a_n =227 \\ \hline \end{array}\)

This is a Arithmetic Sequence.

\(\text{Formula:}~ \boxed{a_n=a_1 + (n-1)d,\text{ with } a_1=2\text{ and }d = 9 }\)

\(\begin{array}{ll} \text{where:}\\ & a_1~ \text{ is the first term, and} \\ & d~ \text{ is the difference between the terms (called the "common difference")} \\ \end{array} \)

\(\begin{array}{|rcll|} \hline 227 &=& 2 + (n-1)9 \\ 227 &=& 2 + 9n-9 \\ 227 &=& 9n-7 \\ 227+7 &=& 9n \\ 234 &=& 9n \\ n &=& \dfrac{234}{9} \\ \mathbf{n} & \mathbf{=} & \mathbf{26} \\ \hline \end{array}\)

...

21*24343535 = 511214235

- PM

The denominator needs to be multiplied too...

- PM

The perimeter of the front of a 10-ounce cereal box is 20 inches. The height is 2 inches more than the width. What are the width and height of the front of the box?

Go try it out yourself. It really works for all m = 6 (mod 9) and n = 0 (mod 9).

- PM