NotThatSmart

first, let's focus in the equation we are given. 

Squaring both sides of the equation, we get that

$(a^2 + 2 + 1/a^2) = 0$

This means that we have $a^2 + 1/a^2 = -2$

Now, let's focus on what we are trying to find. 

Let's notice that 

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

Wait a minute! we already have all the terms needed. We know both values to solve the problem. 

Since the first term is 0, we just have $(0)(-1) = 0$

So 0 is our answer. 

Thanks! :)

First, let's find how many pans of cookies they need to make. 

We just have $216/15=14.4$. since they can't make partial pans, we just round up to 15 pans. 

Now, since they made 15 pans, they're going to need $15 \cdot 3 = 45$ tablespoons of butter to complete the task. 

Divide that by 8, we get $45/8 =5.625$

This rounds up to 6, so they need 6 sticks of butter. 

So 6 is our answer. 

Thanks! :)

First, look at the first equation. Squaring both sides, we have

$a^2 + 2ab + b^2 = 16$

Now, we have two equations to work with. We have

$a^2 + b^2 = 16 - 2ab \\ a^2 + b^2 = 6 + 2ab$

Subtracting the second equation from the first equation, we get

$0 = -10 + 4ab \\ 10 = 4ab \\ ab = 10 / 4 = 5/2$

Now, let's acknowledge something about a^3+b^3. Note that

$a^3 + b^3 = (a + b) ( a^2 + b^2 - ab)$

Wait! we already have all the terms needed to solve the problem!

We have

$\\ (4) ( 6 + 2ab - ab) = \\ (4) ( 6 - ab) = \\ (4) ( 6 - 5/2) = \\ (4) ( 7/2) = \\ 14$

So 14 is our answer. 

Thanks! :)

There are actually multiple parabolas that actually satisfy these conditions. 

However, let's set the origin or $(0, 0)$ as the vertex. 

This would mean the parabola is facing upwards. 

One parabola that wokrs is $y = 2x^2$. 

If that's the case, then we have $2+0+0 = 2$ as our answer. 

So 2 is our answer, i guess. 

Thanks! :)

Let's write a system of equations to solve this problem.. 

Let's set b as bagel. 

Let's let m be muffin.

Let's let t be toast.

We have the system

$2T + 1B = 3.15 \\ 1M + 1B = 3.50 \\ 1T + 2B + 3M = 8.15$

We want to isolate b. From the first 2 equations, we can easily isolate the other two variables to only contain b, We have

$[3.15 - B ] / 2 = T \\ M = 3.50 - B$

Now, we subsitute these two into the third equation. We get

$[ 3.15 - B ] / 2 + 2B + 3 [ 3.50 - B ] = 8.15 \\ [3.15 - B ] + 4B + 6 [ 3.50 -B ] = 16.30 \\ -3B + 3.15 + 21 = 16.30 \\ -3B = 16.30 - 3.15 - 21 \\ -3B = -7.85 \\ B = 7.85 / 3 ≈ $2.6.67$

This is about 262 cents or 261 and 2/3 cents. 

Thanks! :)

First, let's combine all like terms and simplify. We have

$x^2 + 15x + k$ as the simplified version. 

To have a double root means that the descriminant is 0. 

The descriminant is in the form of $b^2-4ac$, so we have the equation

$15^2 - 4 (1)(k) = 0 \\ 225 - 4k = 0\\ 225 = 4k \\ k = 225 / 4$

Thus, our answer is 225/4. 

Thanks! :)

First, let's combine all like terms on the right side of the equation. 

We get the formula

$y=-5x^{2}-6x+10$

Now, it's important to note that the x coordinate of the vertex can be written in the form

$\frac{-b}{2a}$ for parabolas in the form of $y=ax^2+bx+c$

Thus plugging in -5 and -6, we get

$x=6/-10 = -3/5=-0.6$

Plugging this back into the equation, we fins that $y=11.8$

Thus the vertex is at $(-0.6, 11.8)$

So our answer is $(-0.6, 11.8)$

Thanks! :)

First let's combine all like terms and simplify. We have'

$x^{2}+18x+3$

Now, we complete the square for x. Since it is 18x, we add and subtract 81 from the equation. We have

$x^2+18x+81+3x-81\\ (x+9)^2-78$

Thus, the first blank is 1, the second blank is 9, and the third blank is -78. 

Thanks! :)

Let's note something really important first. 

In the problem, it said that $\overline{AB} =4$

Then, it asks us to find the length of AB. 

So, our final answer is $\overline{AB} =4$

So 4 is our answer. 

Thanks! :)