NotThatSmart

Let's first focus on the slope of the line introduced. 

First, note that the slope of a line is in the form \(\frac{y_2-y_1}{x_2-x_1}\)

Thus, plugging in the two points we have, we get

\( [ b^2 - a^2 ] / [ b - a ]=2\)

Now, using the difference of squares thereom, we can simpplify the top to

\( [ (b -a) (b + a) ] / (b -a) = 2\)

The b-a cancels out, leaving us with

\( b + a = 2\)

So 2 is our answer. 

Thanks! :)

Let's use a handy trick to solve this problem. We first have

\(ab -3a + 4b = 131 \)

Now, let's take the coefficients of a and b, which are -3 and 4. We multiply them and take the product and add it to both sides. We get

\(ab - 3a + 4b + (4 * - 3) = 131 + (4 * -3) \)

Now, we simplify and factor the left side of the equation. We get that

\(ab - 3a + 4b - 12 = 119 \\ (a + 4) ( b - 3) = 119 \)

Now, let's focus on the factors of 119. We have

\(1 , 7 , 17, 119\)

Clearly, 7 and 17 will get us the minimal value, so plugging that in, we have

\(( 13 + 4) ( 10 - 3) → |a - b | = |13 - 10 | = 3 \)

so 3 is our final answer. 

Thanks! :)

First, let's zoom in and focus on the equation we are given. Let's square it. We get

\((a^2 - 2 + 1/a^2) = 0\)

This will achieve the equation

\(a^2 - 1/a^2 = -2\)

Next, let's focus on what we are trying to find. We know that

\(a^3 - 1/a^3 = ( a - 1/a) ( a^2 + 1 + 1/a^2)\)

WAIT! The first term in what we must find is 0!

That means our answer is just 0. 

So 0 is our answer!

Thanks! :)

We can write a handy equation to solve this problem. 

First, let's note that every number satsifying the conditions given can be written in the form

\(171x+5\) where x is an integer. 

We set this to equal 1000 to get the largest possible x can be, so we have

\(171x+5 = 1000\)

We are trying to solve for x right now. We have

\(171x=995\)

Dividing both sides, we find that 

\(x=5.81871345029\)

We must round down, so 5 must be our answer. 

So our final answer is 5. 

Thanks! :)

First, let's make some observations about the problem. 

First, let's note that if \(\angle BXA = 90\), then we have \(\angle BXC = 90\)

Triangle BXA is a right triangle, so we can use Pythoagorean Thereom, we simplfy have

\(BC^2 = CX^2 + BX^2 \\ 15^2 = 5^2 + BX^2\\ 225 = 25 + BX^2 \\ 200 = BX^2\)

Square rooting both sides, we finally reach a conclusion. We finally have

\(BX = \sqrt{200} = 10\sqrt 2 \)

So our final answer is 10\sqrt2. 

Thanks! :)

First, let's make some observations about the problem. 

Let's note that triangle ABC is 45-45-90 triangle. 

This means that we can easily find some other lengfths. We have

\(AB = BC = 12/\sqrt 2 = 6\sqrt 2\)

First, let's also note that we als have

\( AD = AD \\ BD = BD \\ AB = BC\)

this finally means that we know that triangle ABD and triangle triangle CBD are congruent triangles. 

Thus, we can finally we have

\([ ABD ] = (1/2) [ABC] = (1/2) ( 1/2) ( 6\sqrt 2)^2 = (1/4) (72) = 18\)

So 18 is our final answer. 

Thanks! :)

It is impossible for thius to happen. 

We have

\(f(x) = \sqrt{x - \sqrt{x}}.\)

No such integers exist. 

Thanks! :)