Let  the number of bird houses sold each month.

The break-even point is where the revenue is the same as the costs:

Solve for :

Therefore, Pets Plus must sell 30 bird houses to break-even.

Let  = the number of birdhouses sold each month.

Substituting in 50 for  gives an answer of 500, so the profit on 50 birdhouses is $500.

Let  = Joey's age and  = George's age.

Then the equation to solve becomes .

Therefore, Joey is 4 years old and George is 12 years old, so the product of their ages is 48.

Let  = 1st even number, = 2nd even number, and  = 3rd even number.

Then the equation to solve becomes .

Thus , so the middle number is 14.

The quantity inside the absolute value brackets must equal either  or , depending on whether the quantity inside the brackets is positive or negative. We therefore have two seperate equations:

To solve the first equation, add 9 to both sides:

Subtract  from both sides:

This is the first solution. Now let's look at the second equation. The distributive law gives us:

Add 9 to both sides:

Add  to both sides:

Divide both sides by 3:

Therefore, is either 4 or 6. 

Statement  does NOT have to be true because  can also equal 4.

Statement  must be true because both 4 and 6 are positive .

Finally, statement  always holds because 4 and 6 are both even. 

The statement expresses the idea that any number is equal to itself. This is the reflexive property of equality.

We need to input 1.5 into our function, then we need to input "a" into our function and set these results equal.

f(a) = f(1.5)

f(a) = -(1.5)² +6(1.5) -5

f(a) = -2.25 + 9 - 5

f(a) = 1.75

-a² + 6a -5 = 1.75

Multiply both sides by 4, so that we can work with only whole numbers coefficients.

-4a² + 24a - 20 = 7

Subtract 7 from both sides.

-4a² + 24a - 27 = 0

Multiply both sides by negative one, just to make more positive coefficients, which are usually easier to work with.

4a² - 24a + 27 = 0

In order to factor this, we need to mutiply the outer coefficients, which gives us 4(27) = 108. We need to think of two numbers that multiply to give us 108, but add to give us -24. These two numbers are -6 and -18. Now we rewrite the equation as:

4a² - 6a -18a + 27 = 0

We can now group the first two terms and the last two terms, and then we can factor.

(4a² - 6a )+(-18a + 27) = 0

2a(2a-3) + -9(2a - 3) = 0

(2a-9)(2a-3) = 0

This means that 2a - 9 =0, or 2a - 3 = 0.

2a - 9 = 0

2a = 9

a = 9/2 = 4.5

2a - 3 = 0

a = 3/2 = 1.5

So a can be either 1.5 or 4.5.

The only answer choice available that could be a is 4.5.

Quadratic equations generally have two answers.  We add 5 to both sides and then divide by 2 to get the quadratic expression on one side of the equation: (x + 1)² = 16.   By taking the square root of both sides we get x + 1 = –4 or x + 1 = 4.  Then we subtract 1 from both sides to get x = –5 or x = 3.

Define the variables to be x = first multiple of three and x + 3 = the next consecutive multiple of 3.

Knowing the product of these two numbers is 54 we get the equation x(x + 3) = 54. To solve this quadratic equation we need to multiply it out and set it to zero then factor it. So x² + 3x – 54 = 0 becomes (x + 9)(x – 6) = 0.  Solving for x we get x = –9 or x = 6 and only the positive number is correct.  So the two numbers are 6 and 9 and their sum is 15.

Generally, quadratic equations have two answers.

First, the equations must be put in standard form: 3x² + 10x + 3 = 0

Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.

Third, check the answer by plugging the answers back into the original equation.