John's Responses in WyzAnt Answers

Since there are no variables under radicals or in denominators, the domain is all real numbers.

Since the equation is a quadratic, the maximum or minimum value will be located at the vertex.

To find the x value of the vertex of the parabola, divide the opposite coefficient of the x term by 2 times the coefficient of the x² term.

x = -300/[2*(-18)]

x = 25/3

Substituting the x value of the vertex into the equation, we can find the y value of the vertex.

y = -18(25/3)² + 300(25/3) + 100

y = -1250 + 2500 + 100

y = 1350

Since the coefficient in front of the x² term is negative, the vertex is the maximum (highest point of the function).  The equation goes down forever on each side.

The range is y ≤ 1350.

I am assuming that you are converting the equation of the circle standard form (possibly for graphing purposes).

First, I am going to arrange the x terms together and the y terms together:

x² - 6x + y² + 8y = -9

Next, I am going to group the x's terms in one parenthesis and the y terms in another parenthesis:

(x² - 6x) + (y² + 8y) = -9

We are going to complete the square in each of the parenthesis.  To complete the square, we need to divde the coefficient of the first power term by 2 and square it.

For the x's, we need to divide -6 by 2 and square the result.  (-6/2)² = (-3)² = 9.  We need to add 9 in the first parenthesis.

For the y's, we need to divide 8 by 2 and squre the result.  (8/2)² = 4² = 16.  We need to add 16 in the second parenthesis.

Remember that the same numbers that are added on the left side of the equal sign must also be added on the right side of the equal sign.

(x² - 6x + 9) + (y² + 8y + 16) = -9 + 9 + 16

We can factor the parenthesis as perfect squares, if we completed the square correctly.  The factors will be the square root of the first term, sign of the middle term, square root of the last term.  The whole factor will be squared.

(x - 3)² + (y + 4)² = -9 + 9 + 16

Simplifying the right side, we get the standard form of a circle

(x -3)² + (y + 4)² = 16

The standard form of a circle is: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

The center of your circle is (3, -4), since 3 is subtracted from x and -4 is subtracted from y.

The radius of the circle is 4, since 4 is the square root of 16.

You need to name variables for plane speed and wind speed.  For example, I will name them:

x = Plane speed

y = Wind speed

Now you can create two equations in two variables

x + y = 158

x - y =112

This is a good problem for elimination method since you already have one equation with positive y and the other with negative y

x + y = 158
x - y = 112
2x    = 270

x = 135

Now substitute the value into either equation to solve for y

x + y - 158

135 + y = 158

y = 23

The first statement tells us the the sum of the tenths digit (t) and the hundredths digit (h) is 13.  The equation for this is:  t + h = 13

The second statement tells us that reversing the digits decreases the number by .45 this means that .th is .45 greater than .ht.  To get .th, we need to multiply t by .1 and h by .01.  From the second statement, we would get the equation:  .1t + .01h = .1h + .01t + .45

To solve this, I would solve the first equation for one of the variables (I will solve for t) and substitute into the second equation.

t + h = 13

t = 13 - h

.1t + .01h = .1h + .01t + .45

.1(13 - h) + .01h = .1h + .01(13 - h) + .45

1.3 - .1h + .01h = .1h + .13 - .01h + .45

1.3 -.09h = .09h + .58

1.3 - .58 = .09h + .09h

.72 = .18h

h = 4

Now that we have one digit, we can substitute the number into the first equation to find the other digit

t + h = 13

t + 4 = 13

t = 9

You need to multiply each term in the first parenthesis by each term in the second parenthesis:

5x²(x² - 3x + 5) + 2x(x² - 3x + 5) + 1(x² - 3x + 5)

5x⁴ - 15x³ + 25x² + 2x³ - 6x² + 10x + x² -3x + 5

Combine like terms to simplify the solution

5x⁴ -13x³ + 20x² + 7x + 5