Find three consecutive odd integers such that the square of the first increased by product of the other two is 224.
Why did the odd integers go on a road trip? Because they wanted to discover the oddity of life!
Now, let's solve this problem while having some fun.
Let's assume the first consecutive odd integer is x. So, the next two consecutive odd integers would be x+2 and x+4.
According to the problem, the square of the first integer increased by the product of the other two is equal to 224. Mathematically, this can be written as:
x^2 + (x+2)(x+4) = 224
Expanding the equation, we have:
x^2 + (x^2 + 6x + 8) = 224
Combining like terms, we get:
2x^2 + 6x + 8 = 224
Subtracting 224 from both sides, we have:
2x^2 + 6x - 216 = 0
Dividing the equation by 2, we simplify it to:
x^2 + 3x - 108 = 0
Now, let's find the factors of -108 that add up to 3. After a bit of clowning around, we find that the factors are 9 and -12.
Therefore, the consecutive odd integers that satisfy the given condition are 9, 11, and 13.
Hope you enjoyed the math humor along the way!
Let's assume the three consecutive odd integers are x, x+2, and x+4.
According to the given condition, the square of the first integer (x) increased by the product of the other two (x+2)(x+4) is equal to 224.
Mathematically, this can be represented as:
x^2 + (x+2)(x+4) = 224.
Expanding the equation:
x^2 + (x^2 + 6x + 8) = 224.
Combining like terms:
2x^2 + 6x + 8 = 224.
Rearranging the equation:
2x^2 + 6x - 216 = 0.
Now, let's solve this quadratic equation. We can either factor it or use the quadratic formula.
Factoring this quadratic equation:
2x^2 + 6x - 216 = 0
2(x^2 + 3x - 108) = 0
2(x - 9)(x + 12) = 0.
From here, we have two possible solutions for x:
1. x - 9 = 0, which gives us x = 9.
2. x + 12 = 0, which gives us x = -12.
However, since we are looking for consecutive odd integers, we will ignore the negative solution. Therefore, x = 9.
So, the three consecutive odd integers are 9, 11, and 13.
Therefore, the three consecutive odd integers, such that the square of the first increased by the product of the other two is 224, are 9, 11, and 13.
To find three consecutive odd integers, let's assign the first odd integer as "x." Since the numbers are consecutive, the next two odd integers can be represented as "x + 2" and "x + 4" respectively.
Now, we can form an equation to solve for the values of x, x + 2, and x + 4. According to the problem statement, the square of the first odd integer increased by the product of the other two is equal to 224. Therefore, we can write the equation as:
x^2 + (x + 2)(x + 4) = 224
To solve this quadratic equation, we need to expand and simplify it:
x^2 + (x^2 + 6x + 8) = 224
2x^2 + 6x + 8 = 224
Rearranging the terms:
2x^2 + 6x - 216 = 0
Next, we need to solve this quadratic equation. We can factor it or use the quadratic formula. In this case, factoring is more straightforward:
2(x^2 + 3x - 108) = 0
Factoring further:
2(x + 12)(x - 9) = 0
Setting each factor equal to zero and solving for x:
x + 12 = 0 or x - 9 = 0
If x + 12 = 0, then x = -12
If x - 9 = 0, then x = 9
These are the two potential values for x. However, we were asked to find the odd integers, and -12 is not odd. Thus, we disregard x = -12.
The odd integers are x = 9, x + 2 = 9 + 2 = 11, and x + 4 = 9 + 4 = 13. So, the three consecutive odd integers are 9, 11, and 13.
i got this so correct omg im so happy
let the middle number be x
then the 1st is x-2
and the 3rd is x+2
(x-2)^2 + x(x+2) = 224
x^2 - 4x + 4 + x^2 + 2x = 224
2x^2 -2x - 220 = 0
x^2 - x - 110 = 0
(x-11)(x+10) = 0
x = 11 or x = -10, but x is to be odd
so
x = 11
and the three numbers are 9, 11, and 13
check:
9^2 + 11(13) = 224 , YEAH!