We are initially presented with a quadratic equation, . To begin we must factor this equation.

The multiples of 15 are (15 and 1) and (3 and 5). The only multiples that add or subtract to  are 3 and 5. Hence we use these as our binomial numbers. . We must now decide on the signs. Because we need to add or subtract 5 and 3 to get to , both signs must be negative: .

From this point we need to switch gears to find solutions to the equation. What numbers would make this equation equal 0?

At this point split the equation into two parts.

 and  and solve.

 and . Both of these numbers inserted into the original equation will produce a result of 0.

Now the question itself is asking for the product of the solutions to the equation, or , which equals 15, therefore 15 is our answer. 

Solving for  yields  and. Solving for  yields  and .

The greatest difference between these two numbers is 14, and 14 squared is 196.

Let  = 1st number

and 

= 2nd number

So the equation to solve becomes

or 

We factor to solve the quadratic equation to get 9 and 12 and their sum is 21.

Let  = 1st odd number and  = 2nd odd number.

So the equation to solve becomes

or

Solving the quadratic equation by factoring gives 9 and 11, so the sum is 20

We can generate an equation for the number of home runs he has hit, x : x² - 50x = 50.  Reordering this, we get : x² - 50x - 50 = 0.  Using the quadratic equation: x = (-b± √(b²-4ac)) / (2a). In this case, a = 1, b = -50, c = -50.  Plugging in these values, we obtain the simplified equation, x = (50±51.96)/2.  Therefore, x = 50.98, -0.98.  Because it doesn't make sense to have a negative number of home runs, x = 50.98, which rounds up to 51 home runs. 

a = –3, b = –6.  The sum of a and b have to be equal to –9, and they have to multiply together to get +18. If a = –3 and b = –6, (–3) + (–6) = (–9) and (–3)(–6) = 18.  

Apply the quadratic formula directly to get [7 ± (49 – 4 * 11 * –8)^0.5]/22, → [7 ± (≈ 20)]/22

So our approximate answers are 27/22 and –13/22, and 27/22 is our answer.

Let  be defined as the lower number, and  as the greater number.

We know that the first number times the second is , so the equation to solve becomes .

Distributing the gives us a polynomial, which we can solve by factoring.

and

The question tells us that the integers are positive; therefore, .

If , and the second number is , then the second number is .

The sum of these numbers is .

4y³ - 4y² = 8y

Divide by y and set equal to zero.

4y² - 4y – 8 = 0

(2y + 2)(2y – 4) = 0

2y + 2 = 0

2y = –2

y = –1

2y – 4 = 0

2y = 4

y = 2

You may use the quadratic formula (where ), which yields two answers,  and .

Since the only solution that appears in the answer list is , we choose .