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to find the maximum value, you have to find the vertex of the quadratic

the formula for the vertex is: \((-\dfrac{b}{2a},f(-\dfrac{b}{2a}))\)

in this case, b=4 and a=-3

so, the vertex is \(\boxed{(\dfrac{2}{3},-\dfrac{17}{3})}\)

The function gets bumped up at every prime values, so the range of p is [2,5] U [8,9].

The difference is 26^3*10^3 - 26^2*10^4 = 10816000.

CDEFG is 384 cm

We simplify all of these expressions.

A:

\(A=\frac{2^{\frac{1}{2}}}{4^{\frac{1}{6}}}\\ A=\frac{2^{\frac{1}{2}}}{(2^2)^{\frac{1}{6}}}\\ A=\frac{2^{\frac{1}{2}}}{2^{\frac{1}{4}}}\\ A=2^\frac{1}{4}\)

B:

\(B=\sqrt[12]{128}\\ B=\sqrt[12]{2^7}\\ B=2^\frac{7}{12}\)

C:

\(C=(\frac{1}{8^\frac{1}{5}})^2\\ C=(\frac{1}{(2^3)^\frac{1}{5}})^2\\ C=(\frac{1}{2^\frac{3}{5}})^2\\ C=(2^{-\frac{3}{5}})^2\\ C=(2^{-{\frac{6}{5}}})\\ \)

D:

\(D=\sqrt{\frac{4^{-1}}{2^{-1}\cdot8^{-1}}}\\ D=\sqrt{\frac{(2^2)^{-1}}{2^{-1}\cdot(2^3)^{-1}}}\\ D=\sqrt{\frac{2^{-2}}{2^{-4}}}\\ D=2\)

E:

\(E=\sqrt[3]{2^\frac{1}{2}\cdot4^-{\frac{1}{4}}}\\ E=\sqrt[3]{2^\frac{1}{2}\cdot(2^2)^-{\frac{1}{4}}}\\ E=\sqrt[3]{2^\frac{1}{2}\cdot2^-{\frac{1}{2}}}\\ E=\sqrt[3]{1}\\ E=1\)

We then order them: C, E, A, B, D.

We complete the square to get \(y=-2(y-3)^2+3\). The vertex is (h, k). Therfore the vertex of the parabola \(x = -2y^2 + 12y - 15\ \text {is} \ \boxed{(3, 3)}.\)

The line perpendicular to y=2x+5 must have the slope -1/2. We then have that the perpendicular line's equation is y=-1/2x+b. We plug in the given point (5, 5) to solve for b.

y=-1/2x+b

5=-1/2(5)+b

5=-5/2+b

b=15/2

So the equation of this line is y=-1/2x+15/2. To find the intersection point of it and y=2x+5, we set -1/2x+15/2 and 2x+5 to be equal to each other.

-1/2x+15/2=2x+5

5/2x=5/2

x=1

Plug this value into either of the equations for the lines we have. y=2x+5 will be easier to work with.

y=2x+5

y=2(1)+5

y=7

Therefore the intersection point is (1, 7).

Let x be the cost of a brownie and y the cost of a cookie. The amount of money Melvin spends is 4x+3y. The amount of money Lily spends is 2x+16y, which is double of 4x+3y(the amount of money Melvin spent). We then form an equation.

2(4x+3y)=2x+16y

8x+6y=2x+16y

6x=10y

x=5/3y

Therefore a brownie is 5/3 times as expensive as a cookie.

we get (9^x)-2*(3^x)=63 by adding 63 to both sides and then we use elimination.Finally we get x=2 so the only way for x is 2

Well, what happened to your pictures??

we simplify sqrt2014 to get sqrt(2*19*53) and we make it sqrt(8*19*53)=sqrt8056 and finally it is sqrt(91+sqrt8056) which is about 13.44452389… but I don't know how to simplify it

I'd use a calculator; it doesn't seem to have a "nice" answer.

Solve:         9^x - 2·3^x - 63  =  0

Rewrite:  (3²)^x - 2·3^x - 63  =  0

                  3^2x - 2·3^x - 63  =  0

Factor:      (3^x - 9)(3^x + 7)  =  0

Either:     3^x  =  9     --->     x  =  2

or:           3^x  =  -7    --->    (no real solution)    

Hint:Use complementary

Well find all the ways then subtract the ways that you don't want

(2x + 3) / (x² - 6x + 8)  =  A / (x - 2) + B / (x- 4)

Since  x² - 6x + 8  can be factored as  (x - 2)(x - 4),

multiply each term by  (x - 2)(x - 4)  to get:     2x + 3  =  A(x - 4) + B(x - 2)

                                                                        2x + 3  =  Ax - 4A + Bx - 2B

Set the x-terms equal to each other:  2x  =  Ax + Bx     --->    2x  =  (A + B)x     --->   A + B  =  2

Set the constants equal to each other:  3  =  -4A - 2B   --->   4A + 2 B  =  -3

Solve this set of two equations:     A +   B  =    2     to get your answer.

                                                     4A + 2B  =  -3  

If you had the problem       :  x² - 21x - 100  =  0

you would factor it this way:  (x - 25)(x + 4)  =  0

and the solutions are:  25,  -4

Since you have:     x⁴ - 21x² - 100  =  0

factor it as:           (x² - 25)(x² + 4)  =  0

This can be further factored as:                                            (x - 5)(x + 5)(x² + 4)  =  0

If you have complex numbers you can factor again:   (x - 5)(x + 5)(x - 2i)(x + 2i)  =  0

and the solutions are:  5,  -5,  2i,  -2i

If you don't have complex numbers, stop with:  (x - 5)(x + 5)(x² + 4)  =  0

and the solutions are:  5,  -5

My computer says there are 375 ways to get $20 from 4 people.

1)  Rewrite  3x² + 12x + 4  into the form  a(x - h)2 + k

     3x² + 12x + 4

Separate the x-terms from the constant:  (3x² + 12x) + 4

Factor out the 3:                                        3(x² + 4x) + 4

Complete the square:                                3(x² + 4x + 4) - 12 + 4 * (see below)

Factor:                                                       3(x + 2)² - 8

Pick out a, h, and K:      a = 3     h = -2     k = -8

*Completing the square:  divide the coefficient of the x-term (4x) by 2 and square the result:

               4 / 2  =  2     --->     2² = 4

  place this number within the parentheses

  because you have entered a 4 in the parentheses and the parentheses have a '3' before it,

     you have added a '12', so you'll need to subtract a 12.

2)  x - y² + 8y  =  13

     Rearrange:                             - y² + 8y  =  13 - x

     Multiply everything by -1:          y² - 8y  =  x - 13

     Complete the squre:         y² - 8y + 16  =  x - 13 + 16

     Factor:                                      (y - 4)²  =  x + 3

     Vertex:       x = -3     y = 4

3)  Like number 2.

I mean yay because our super answerer is back