The product of two consecutive integers is 462. Find the integers?

Question

How Do i find the integers?

Answer 1

Let x be the first integer.  Then the second integer is x + 1.  Now we multiply them together to get this:

x (x +1) = 462

x² + x = 462

Now move the constant on the right to the left side, and set the equation equal to 0:

  

x² + x - 462 = 0.

Now factor the equation:

(x + 21)(x - 20) = 0..

Set each factor equal to 0 and solve for x for each factor:

x + 21 = 0 ----> x = -21

x - 20 = 0 ----> x = 20.

This leads us to two intriguing possibilities.  The problem does not say that the integers have to be positive or negative, so we can actually use the positive and negative value of 20 and 21.  

If x = 20, then x + 1 = 20 + 1 = 21, and 20 x 21 = 462.  So this solution works.

If x = -21, then x + 1 = -21 + 1 = -20, and -21 x -20 = 462.  So this solution works as well.

So here are the integers: x = -20 and -21, and x = 20 and 21.

Answer 2

Find the closest numbers to 462 that are perfect squares. The closest are 441 (21 x 21) and 484 (22 x 22). That should give you the help you need.

Answer 3

x = first integer

x + 1 = second integer (consecutive)

x * (x + 1) = 462

x² + x = 462

Convert the left side into a perfect square by adding some unknown value (A). To keep things in balance, we have to add A to both sides.

x² + x + A = 462 + A

To calculate A, we first take 1/2 of the coefficient of the first degree term (x).

That coefficient is 1.

And 1/2 of 1 = 1/2.

Then we square that to get 1/4.

This is the value of A.

x² + x + 1/4 = 462 + 1/4

Now that we have a perfect square on the left, we can factor it.

(x + 1/2)(x + 1/2) = 462.25

(x + 1/2)² = 462.25

take the square root of both sides

x + 1/2 = 21.5

x = 21.5 - 0.5

x = 21

and

x + 1 = 22

Since 21 * 22 = 462, the first integer is 21 and the second consecutive integer is 22.

Answer 4

Here's how I would do it.

Let x = the first integer.

x + 1 = the second integer

x (x+1) = 462

x^2 + x - 462 = 0

Then, by guessing and checking or quadratic formula, you can factor that to:

(x + 22) (x - 21) = 0

Therefore, x is either -22 or 21.

You then plug that into the equation to get that there are two possible solutions.

The integers are either 21 and 22 or -22 and -21.

Answer 5

A product is a multiplication.

Choose one integer x.  And  its successor y = x + 1

x * y = 462.

x * (x + 1) = 462

Using quadratic formula, solve for x.

x = 21, y = 22

21 * 22 = 462

Answer 6

Get in the ball park by squaring even tens - for example 10x10=100 & 20x20=400 & 30x30=900.  Now you know the answer is between 20 and 30 and since 400 is the closest to 462 start with 20.  Since you know the integers are consecutive you could start with 20x21 but you also know the last digit is a 2 so whatever you multiply together must end with a 2 so you know it isn't 20x21 because that would end in a zero (420).  Try the next 2 consecutive integers which are 21 & 22 and 21x22=462 and you don't even need a calculator to come up with the answer!

Answer 7

I thought about you problem this way.

It is almost the same as asking: "The product of two equal numbers is 462. Find the number".

So we're thinking about squares and square roots.

The square root of 462 is about 21.49, so it's pretty obvious that the answer to your question is 21 multiplied by 22.