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Move all of the terms to the left side and set equal to zero. Then set each factor equal to zero. Which any number multiplied by zero is zero

1 X 0 = 0

15,000X 0 = 0

120,000,000 X 0 = 0 

Etc.

Find all integers n such that the quadratic 7x^2 + nx - 30  can be expressed as the product of two linear factors with integer coefficients.

Hello Guest!

\(7x^2 + nx - 30 =0\)

\(x =\dfrac{-n\pm \sqrt{n^2+4\cdot 7\cdot 30}}{2\cdot 7}\\ x=-\dfrac{n}{14}\pm \dfrac{ \sqrt{n^2+840}}{14}\)

\(\{n^2+840)\}\subset \{square\ numbers \}\\ \{{\color{blue}11^2}+840)\}\subset \{square\ numbers \}\ determined\ graphically \\\)

\(x =\dfrac{-n\pm \sqrt{n^2+4\cdot 7\cdot 30}}{14}\\ x =\dfrac{-11\pm \sqrt{11^2+840}}{14}\\ x=\dfrac{-11\pm 31}{14}\\ \color{blue }x=-3\\ \)

\(\color{blue }7x^2 + 11x - 30 =0\\ 7\cdot (-3)^2 + 11\cdot (-3)- 30 =0\\\)

Help! I don't know what to do next.

  ! asinus

Here's a non-calculus answer.

\(\displaystyle x-3=2\sin(2t)\\y-4=2\cos(2t).\)

Squaring and adding,

\(\displaystyle (x-3)^{2}+(y-4)^{2}=2^{2},\)

so the particle is moving on a circle centre(3, 4) radius 2.

At t = 0, the particle is at the point (3, 6), (the top of the circle), and as t increases sin(2t) increases so x increases,

implying that the particle is moving clockwise around the circle.

The particle will be at x = 3 when sin(2t) = 0, that is, when 2t = 0, pi, 2pi, ..., t = 0. pi/2, pi, ... .

At t = 0,     x = 3, y = 6.

At t = pi/2, x = 3, y = 2.

At t = pi,    x = 3, y = 6, etc.

So, it takes pi seconds(?) to go once round, and since the circumference is 4pi it follows that its speed is 4.

As follows:

You will need to know the specific heat capacity of ice to go from -30C to 0C; then the latent heat of ice/water to turn the ice into water, then the specific heat capacity of water to go from 0C to 100C, then the latent heat of water/steam to turn the water into steam, then the specific heat capacity of steam to go from 100C to 40C. 

15 3/4   - 2 cups saved = 13 3/4 cups available for bread

13 3/4   /   5 1/2     =  55/4   /   11/2    =     55/4  * 2/11 = 110/44 = 110/44 = 2.5 breads

C would have modulus of sqrt 10     to correspond to answer found in part A of the Question

x⁵  =  - x³

x⁵ + x³  =  0

x³(x² + 1)  =  0

If you are restricted to real numbers, the only solution for x is 0.

If you can use compex numbers:

x³(x² + 1)  =  0

Either  x = 0  or  x² + 1  =  0

If  x² + 1  =  0,  then  x  =  i  or  x  = -i,  giving you three solutions:  x = 0  or  x = i  or  x = -i.

In a card game played with number cards from 1 to 9, you win based on the total value of your cards. If you have five cards that total 5, you beat someone with cards that total 25. How could that be?     

There is only one way that 5 cards can total 5 points.  

There are several ways that 5 cards can total 25 points. 

So the 5 points hand is more rare, therefore more valuable.  

It should be care

does anyone know what the answers are???

i got 2145 for a and 1742 for b lemme know if this is right

I would be happy to give you hints but I haven't got a clue.

Maybe appeal to Heureka....   But, as a guest, it is almost impossible for you to ask anyone directly.

I think

\(\frac{dx}{dt}=2cos(2t)\\ \frac{dy}{dt}=-2sin(2t)\\~\\ \frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{dt}{dx}=-2sin(2x)*\frac{1}{2cos(2x)}=-tan(2x)\)

I admit it. I don't know. 

74 x  5%    =   74 x .05 = 3.70 dollars

ok, not thay hard. we can make a variable.

say x = pieces of candy Sam has.

so Kabir has 5x

and the total is x + 5x = 6x

then we know that 5x - 18 = 3x        (the 3x is from 6x/2)

\(5x-18=3x \)

\(5x=3x+18\)

\(2x=18\)

\(x=9\)

so the total is 6x, 6 * 9 = 54.

You can think of the square without the center piece. This figure would consist of \(8\) squares, \(4\) being shaded or \(\frac{1}{2}\) shaded. We can think of the center piece of this figure as \(8\) squares again with \(4\) shaded. Each time you count the center piece with \(8\) squares, you have \(\frac{1}{2}\) shaded and \(\frac{1}{2}\) unshaded. This pattern continues forever. Thus the answer is \(\frac{1}{2}.\)

I found this:

https://web2.0calc.com/questions/in-the-sequence-1-2-2-4-8-32-256-each-term-starting-from-the-third-term-is-the-product-of-the-two-terms-before-it-for-example-the-seventh

ah nvm, read that wrong

I believe you use power of a point for this problem. 

https://artofproblemsolving.com/wiki/index.php/Power_of_a_Point_Theorem

Here's a nice article to read if you don't know it. 

6^2 = x*(x+5)

36 = x^2 + 5x

x = 4

I hope that's correct :))

=^._.^=

Sorry, I made a mistake.

May I ask how did you come up with 2^30?

2^30