First you must find the solutions to the equation. This can be done either by using the quadratic formula or by simply finding two numbers whose sum is 5 and whose product is -24 and factoring the equation into (x + 8)(x – 3) = 0. The solutions to the equation are therefore -8 and 3, giving a sum of -5.

Notice that when the ball is at ground level, the height is zero. Setting s (t) equal to zero and solving for t will then give the times when the ball is at the ground.

–t² + 4t =0

t(4 – t) = 0

t = 0, t = 4

The ball returns to the ground after 4 seconds.

Define varables as  = the first number and  = the second number.  The product of the numbers is .  Solve the resulting quadratic equation by factoring and setting each factor to zero.  The numbers are 12 and 15 and the sum is 27.

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.