the product of two even consecutive integers is 224. find the integers
14 and 16.
how do u figure that out?
Just try them quickly, there's a more formal method, but it'd be much more complicated for such a simple problem.
They want only even numbers. You can quickly see 10 x 12 = 120 is smaller than 224 and 18 x 20 = 360 is too big.
You're left with 12 x 14, 14 x 16, etc. Turns out 14 x 16 works.
Anyway, here's the formal method :
Let "Y" be an even number.
We want Y x (Y+2)= 224
We get Y^2 +2y -224=0
And the solution for this equation is Y=14. So the answer is Y and Y+2, 14 and 16.
To find the two even consecutive integers whose product is 224, we can set up an equation.
Let's assume the first even integer is x. Since the integers are consecutive, the second even integer would be (x + 2) since we are dealing with even numbers.
The equation representing the problem is: x * (x + 2) = 224.
To solve this equation, we need to determine the values of x and (x + 2) that satisfy it. Here's how to do it:
1. Expand the equation: x^2 + 2x = 224.
2. Set the equation equal to zero by subtracting 224 from both sides: x^2 + 2x - 224 = 0.
3. Factor the quadratic equation or use the quadratic formula to find the values of x that satisfy the equation.
Factoring method:
By factoring or using trial and error, the factors of -224 that add up to 2 are -14 and 16.
So the factored form of the equation is: (x - 14)(x + 16) = 0.
Quadratic formula method:
You can also solve the equation using the quadratic formula, which is: x = (-b ± sqrt(b^2 - 4ac)) / (2a).
For our equation, a = 1, b = 2, and c = -224.
By substituting these values into the formula, you can find the solutions for x.
Regardless of the method you choose, the solutions for x will be x = -16 and x = 14.
Since the problem states that the integers are even, we only consider the positive solution: x = 14.
Therefore, the two even consecutive integers are 14 and 16.