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I got it and its 10901 but thanks!!!

Hello, Lightning! 

The quadratic formula is a formula that solves for the roots of any quadratic. Let's apply it to the quadratic \(4x^2+7x+k\).

Obviously, we do not know what k is, but we do know that the roots of the quadratic with the unknown k are \(x_{1,2}=\frac{-7\pm i\sqrt{15}}{8}\).

Do you think you can take it on from here? 

I think it is 11 0 11 = 1,001 x 11

i still dont get it, yes i seen your hint but i still have no clue what im suppose to do :/

Hey Lightning, \(a=4\)

\(4x^2+12x+9\\ =(2x+3)^2\)

I used: \((x+y)^2=x^2+2xy+y^2\)

I saw the nine, so I know \(y=3\)

\(2xy=12\), and \(y=3\), so \(x=2\)

\(2^2=4, a=4\)

I hope this helped,

Gavin

This is factorable:

\(-x^2-14x-24=0\\ =(−x−2)(x+12)=0\\ x_1=-2;x_2=-12\)

I'm assuming that there is a equal zero at the end of the equation. 

I hope this helped,

Gavin

Hey Guest,

I have slept on it, and finished the solution. 

First, we use \(x = \tan θ\)

We can rewrite the desired integral as \(\int_0^{\pi/4}\log(\tan(\theta)+1)d\theta\)

Also:

\(\log(\tan(θ) + 1) = \log(\sin(θ) + \cos(θ)) − \log(\cos(θ))\)

Note that: \(\sin(θ) + \cos(θ) = \sqrt2 \cos(π/4 − θ).\)

Rewriting the integral, we get:

\(\frac12\log(2) + \log(\cos(π/4 − θ)) − \log(\cos(θ)).\)

Over the interval \([0, π/4]\), the integrals of \(\log(\cos(θ))\ \text{and}\ \log(\cos(π/4 − θ)) \) are equal, so their contributions cancel out. The answer you seek is \( π \log(2)/8. \)

Gavin

Edit: I have done some research, and googled Putnam competition. 

This problem is A5 of the 66th annual competition. 

The full solution is here: http://math.hawaii.edu/home/pdf/putnam/2005.pdf.

I've seen these questions before. If these are homework problems, this is not a homework site!

Probability: the amount you want/total.

Since there are a total of 400 people, and you want the probability that the person has blood type B, we need to find the total number of people who have blood type B. In the problem it says 120 people have blood type B, so plugging the values into our formula, we have: \(\frac{120}{400}=\frac{12}{40}=\boxed{{\frac{3}{10}}}\)

See the solution here, especially the last one #11 by MaxWong:

https://web2.0calc.com/questions/help-plzzzz_3

12^15 = 15407021574586368 = 2^30×3^15 (45 prime factors, 2 distinct). It also has 496 divisors.

I already gave you a hint, here: https://web2.0calc.com/questions/2-questions-triangle-jkl-and-triangle-mno-are-similar

Please don't double post!

1.

 lcm[12,t]^3=(12t)^2

Lcm[12^3, t^3] = 144t^2

Lcm[1,728, t^3] = 144t^2

The smallest positive t = 18, so that:

Lcm[12^3, 18^3] = 12^2 * 18^2

UPDATE: You don't need to fully answer this question, just please give me an explanation on what independant and dependant means 

The reason Gavin did it his way is that most times this would not work for a table of values.

You would not be able to pick the exact right x value to put into the table. BUT this one has been chosen so that it is easy.

f(x)=x^2-6x+8  this is a parabola, maybe you are meant to know that

g(x)=x-2   this is a line and maybe you are meant to know that as well.

There will be at most two points of intersection between a line and a parabola, lets see if we can find them

f(x)=x^2-6x+8

g(x)=x-2

I can see that for x=2     f(2)=0 and g(2)=0 

so f(x)=g(x) when x=2           So the first solution is x=2

Now continue the tables and see if you can find another match :)

Thanks os much for this, but what would it look like if we did a table of values? 

I have deleted the bit where you gave the final answer Tertre.

Gavin's initial answer was very good.

But Gavin I was surprised that you use k instead of m for the gradient.

To my knowledge m is (almost) universally used for gradient 

I have seen a different letter used only in relation to statistics.

Maybe some countries are different.......

Yeah, I know, I'm just giving them information.

y=mx+b

Since m is the slope.       

And, b, which is the y-intercept

Hey tertre!

You are absolutely correct if someone wants to find the equation of the line with two given points. 

However, the equation of the line is already given to us, we just need to identify the slope. 

:)

-15<3 n<=6      divide all by 3

- 5 < n <= 2

n <= 2              (assuming n is positive)

n = +,- 2

n = +, -1

n = - 4

n = - 3

n = 0

Multiplicaion!

Aw man, i forgot this one

In an equation of a line, we have y, x, the slope (any variable) and the y-intercept. 

You generally see the equation like this: y=mx+b.

 Still with me?

Next, to find the slope, you have to subtract \(y_1\) from \(y_2\) , to get \(y_2-y_1\)

And, then we have to subtract both x-corrdinates-\(x_2-x_1\)

So, the formula to find the slope is \(\frac{y_2-y_1}{x_2-x_1}\) , where \(y_1\) is the first y-coordinate,  \(y_2\) is the second y-coordinate, \(x_1\) is the first x-coordinate, and \(x_2\) is the second x-coordinate.

I know this is a lot of information, question me if you need any help!

This is an equation for a line in slope-intercept form. 

In the line: \(y=kx+b\), the slope is \(k\) and the y-intercept is \(b\), make sure you understand why. :)

1. What do similar triangles mean? Similar Triangles are triangles that have proportionate angles and sides. 

In triangle 1, we have the height as 12cm, and the base as 3.5 cm.

In triangle 2, we have the height as x cm and the base as 14 cm.

So, we need to find an equal proportion between 3.5 cm and 14cm. 

Tell me what you think, our next move should be.