Two positive integers are in ratio 1:3. If their sum is added to their product, the result is 224. Find the integers.

On my own, I came up with the formula 224 = x + 3 x + 3x^2 but it's not really working out. Could someone please help me?

Your formula is correct.

Assuming you add up the 3x and x to form 4x, you have to use the general solution of a quadratic equation. Do you know how to use it? =)
By the way, x should be 8 if you use it right.

no, I don't think I do. I tried completing the square and I'm just getting really weird numbers. :(

Okay, we'll do it by completing the square. :)

From 3x^2 + 4x = 224,
x^2 + 4/3 x = 224/3
x^2 + 4/3x + 4/9 = 224/3 + 4/9
(x+2/3)^2 = 75/1/9
x + 2/3 = square root of 75/1/9
x + 2/3 = 8/2/3 or -8/2/3
x = 8 or -9/1/3( Not applicable as positive integer cannot be a negative fraction)
Hence, x = 8.
There you go! :)

thanks! :D

You're welcome ^^ :)

To solve this problem, let's first set up the equation based on the given information.

Let's assume the two positive integers are x and 3x, as they are in a ratio of 1:3.

The sum of the two integers is x + 3x, and their product is x * 3x, which can be written as 3x^2.

According to the problem, the sum of the integers added to their product is 224, so we can set up the following equation:

(x + 3x) + (x * 3x) = 224

Simplifying this equation will lead us to the answer.

Now, let's solve the equation step by step:

Step 1: Combine like terms on the left side of the equation.

4x + 3x^2 = 224

Step 2: Rearrange the equation to make it a quadratic equation.

3x^2 + 4x - 224 = 0

Step 3: Factor the quadratic equation.

Factoring may not always work for every quadratic equation, but in this case, the equation can be factored.

(3x - 14)(x + 16) = 0

Step 4: Set each factor equal to zero and solve for x.

3x - 14 = 0 or x + 16 = 0

Solving the first equation:

3x = 14
x = 14/3 (approximately 4.67)

Solving the second equation:

x = -16 (which is not a positive integer, so we discard this solution since the problem states that the integers are positive)

Therefore, the positive integer is approximately 4.67.

Since the problem states that the integers must be positive, we need to find the second integer, which is 3 times the value of the first integer.

3x = 3 * (14/3)
3x = 42/3
x = 14/3 (approximately 4.67)

Therefore, the two integers are approximately 4.67 and 14 (approximately 4 and 14).