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! :)