NotThatSmart

Let's give this problem a shot. 

Now, we know that \((a+b)^2 = a^2+2ab+b^2\)

From the problem, we ALSO know that \((a+b)^2 = 1^2=1\)

Thus, we have the equation \(a^2+2ab+b^2=1\)

Since the problem gives us the value of \(a^2+b^2=2\), we can easily figure out what ab is. 

We have

\(2ab+2=1\\ 2ab=-1\\ ab=-1/2\)

The value of ab will come in handy later. 

Now, let's focus on what we must find. a^3+b^3. From a handy equation, we know that

\(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)

Wait! We already know all the values needed to solve the problem. Plugging in 1, 2, and -1/2, we get

\(a^3+b^2=(1)(2-(-1/2)) = 2+1/2 = 5/2\)

Thus, our final answer is 5/2. 

Thanks! :)

Let's make some observations of the problem. 

First, note that triangle ABC is a 45-45-90 right triangle. This means that \(AB = BC = 12/\sqrt 2 = 6\sqrt 2\)

Also, we have that

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

This means that triangles  ABD  and CBD  are congruent

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

So our aswer is 18. 

We must take the highest powers and add them together. 

\(5+6=11\)

So the answer is 11. 

Thanks! :)

Ok, we can go one by one to solve these problems. 

Hoever, let's note that if we have f(x) = x, then we set the right side of the function to equal x.

Then, we see if the x value satisfies the conditions given. 

First, we have \(f(x) = 2x + 8\). Setting this to equal x, we find that \(x = 2x + 8\)

Solving for x, we find that \(x=-8\). This doesn't satisfy the condition \(1 \le x \le 2\), so it is not a valid solution. 

Next, we have \(f(x) = 13 - 5x\). Setting this to equal x, we have \(x = 13 - 5x\)

Finding the value of x, we get \(x=13/6\). This satisfies the inerval, so it is a solution. 

Third, we have \(f(x) = 20 - 14x\). Since this equals x, we can write \(x= 20 - 14x\)

Trying to find x, we have that \(x=20/15=4/3\). This does NOT satisfy the interval, so it is not a solution. 

Lastly, we have \(f(x) = 40 + 5x\). setting this function to equal x, we have \(x = 40 + 5x\)

Solving for x, we get \(x=-10\), which is invalid. 

Thus, the only value that works is 13/6. 

So our answer is 13/6. 

Thanks! :)

Ok, so we start off with the equation

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

I know this is random, but let'st ake the product of the coefficients on a,b.....add this product to  both sides

So this gets us

\(ab - 3a + 4b + (4 * - 3) = 131 + (4 * -3) \\ ab - 3a + 4b - 12 = 119\)

This eventually achieves the equation

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

Now, the factors of 119 are \(1,17,119\)

Since we want the minimum, let's use 7 and 17. We have

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

Thus, our final answer is 3. 

Thanks! :)

Let's note that If AD  = AB  then angle  ABD  =  angle ADB

This means that \(\angle ABD  = (180 - 17)  / 2   =  81.5°  \)

So 81.5 is the answer. 

Thanks! :)