proyaop

First we open the parenthesis.

\(4(-x^2 +5x + 84)\)

Then we open up parenthesis again.

\(-4x^2 + 20x + 336\)

When x = 0, the value is 336.

When x = 1, the value is 352.

When x = 2, the value is 360.

When x = 3, the value is also 360.

But we see when x = 2.5, the value is 361. 

If x = 2.4, the value is 360.96

If x = 2.6, the value is 360.96.

So that means the medium value for x is 2.5, which makes the expression the maximum value.

For the maximum value of the expression, it is 361.

That also works Tiggsy 

First things first, we set the expression to be x^2.'

Next we can say 4b + 9 = x, and 6b + 13 = x.

Then using the transitive property of equality, 4b + 9 = 6b + 13.

b = -2

But the question is asking for a positive integer for b, and that does not have a valid way to quickly solve it. I checked this problem on MATHCOUNTS and it asks for a whole number/integer. so the answer is b = -2

To get the solutions, we must first apply the quadratic formula to the equation above. 

The quadratic formula is: \(-b {+\over} \sqrt{b^2 - 4ac}\over2a\) where \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant.

Now to apply it.

\(31 {+\over} \sqrt{961 - 1008}\over12\) 

Now we have the two solutions.

Here they are:

\(31 + i\sqrt{47}\over12\) , \(31 - i\sqrt{47}\over12\)

Let us simplify first by opening the parenthesis.

\(3x - 12 + 2x^2 - 2x + 14 - 5x + 5 + 8x^2 - 8x + 24 - 4x + 20\)

Now we combine like terms:

\(10x^2 - 16x + 51\)

Now we can see that the constant is 51.

 tedious work!

First to find the equation of the line. The y-intercept rests at (0,1) so the b term in y = mx + b is 1. The slope is -1 when you look at the rate of change in y/x. Plugging in the values, you get the equation of the line.

The equation for this line is \(y = -x + 1\), but because the shaded region is ABOVE the x-axis and the line, and because this question is asking for a linear inequality, then the linear inequality for this question is:

\(y > -x + 1\)

First let's consider the person who had 12$ to start with. If he had 2$ left, that means he spent 5/6th of his money, which you can cross of that option now. You can also cross out the 12$ option for the quantity people start off with. Taking a look at the clues, Alexander is the only person not included, so this person is Alexander. Then Alexander spent 10$ at the mall, and has 2$ left.

Now we consider Jordan. If we know jordan spent 24$, then taking a look at the options for how much people started with, we can see Jordan spent either 1/3 of his money or 1/2 of his money. 11/12 and 1/12 would not be valid options or else Jordan would have too much or too little money in the list of money you start off with. But we are also given Jordan spent LESS than 1/2 of his money, meaning Jordan could only have spent 1/3 of his money. 24 x 3 = 72, so Jordan had 72$ to begin with. Then Jordan spent 24$ at the mall and he has 48$ left. 

Now we consider Justin. If Justin had 24$ left after spending money, then Justin either spent 1/12, 11/12, or 1/2 of his money. But if he spent 1/12 or 11/12 of his money, he would either have too much or too little money to begin with. Then the last option is 1/2, so he spent 1/2 of his money, meaning Justin started off with 48$. Then Justin spent 24$ at the mall and he has 24$ left.

Now we consider Benjamin. If Benjamin spent 8$ at the mall, that means he either spent 1/12 or 11/12 of his money. If he spent 11/12 of his money, that means he didn't have a whole number amount of money to begin with, and that isn't an option on the list. Therefore he spent 1/12 of his money, meaning Benjamin started off with 96$. Then Benjamin spent 8$ at the mall and he has 88$ left.

Now we consider Zachary. We know he has spent 11/12 of his money because that is the last option. The last option for money to start off with is 60$, so we now know Zachary spent 11/12 of his money and had 60$ to start. Then Zachary spent 55$ at the mall, and has 5$ left.

 Good question!

The answer would then be \(3 + (-4) + 24\) which is 23.

The first step is to combine like terms in the original expression in the question to make the quadratic clearer:

\(x^2 + 6x + 14\)

Now we must open the parenthesis for (x + u)^2.

\((x + u)^2\) = \(x^2 + 2xu + u^2\)

\(u\) is a constant so \(u^2\) is also a constant, meaning that \(2xu\) is the only term that can equal \(6x\) in the expression in the question. 

Therefore the constant \(u = 3\). 

If \(u\) is 3, then \(u^2\) would be 9, making the new expression: 

\(ax^2 + 6ax + 9a + v\)

(don't forget about the variable \(a\) that multiplies the outcome of opening the parenthesis for (x + u)^2)

If you take a look at the expression the problem gave to us, the coefficient of \(x^2\) was 1, so that means the constant \(a\) is also 1.

The new expression would look like this and we can get the last constant we need:

\(x^2 + 6x + 9 + v\)

Now we see there is a difference of 5 from the expression above and the expression in the problem, so \(v = 5\).

Therefore the answer for this questions is:

\(1(x + 3)^2 + 5\)

where \(a = 1\), \(u = 3\), and \(v = 5\)

The first step is to consider some triangles.

Once we find x, we can see that there are two congruent triangles with hypotenuse length \(10\sqrt{3}\).

One of the legs of this triangle will be y1 - y2, which is 14.

Using the pythagorean theorem, we can find the distance between the two x coordinates. 

\(\sqrt{(10\sqrt{3})^2 - 14^2}\)

This simplifies to:

\(\sqrt{104}\)

This simplifies to:

\(2\sqrt{26}\)

This means the x coordinate is \(2\sqrt{26}\) units away from the x coordinate of (4,7), and then we have these 2 possible values of x:

\(4 + 2\sqrt{26}\) and \(4 - 2\sqrt{26}\)

Since the question is looking for the sum of the possible values of x, then we just add the two possible values together:

\(4 + 2\sqrt{26} + 4 - 2\sqrt{26}\)

This simplifies to 8.

Therefore, the sum of all possible values of x is 8.