Find two consecutive odd integers such that the square of the first, added to 3 times the second, is 24.
To solve this problem, let's represent the first odd integer as x and the second odd integer as x + 2 (since they are consecutive odd integers).
The square of the first integer, x, is x^2.
Three times the second integer, x + 2, is 3(x + 2).
According to the problem, the sum of the square of the first integer and three times the second integer is 24. So we can create the equation:
x^2 + 3(x + 2) = 24
Now, let's solve this equation step by step:
1. Expand the equation:
x^2 + 3x + 6 = 24
2. Move 24 to the right side by subtracting 24 from both sides:
x^2 + 3x + 6 - 24 = 0
3. Simplify:
x^2 + 3x - 18 = 0
Now, we have a quadratic equation that we can solve. To solve it, we can use factoring, quadratic formula, or completing the square method. Let's use the quadratic formula:
4. Use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the equation:
x = (-3 ± √(3^2 - 4(1)(-18))) / (2(1))
Simplifying further:
x = (-3 ± √(9 + 72)) / 2
x = (-3 ± √81) / 2
x = (-3 ± 9) / 2
Using the plus-minus symbol, we can find two potential values for x:
1) x = (-3 + 9) / 2 = 6/2 = 3
2) x = (-3 - 9) / 2 = -12/2 = -6
Since we are looking for consecutive odd integers, we can discard the negative value, -6, and conclude that the first odd integer is 3.
The second odd integer is obtained by adding 2 to the first odd integer:
x + 2 = 3 + 2 = 5
Therefore, the two consecutive odd integers whose square of the first added to 3 times the second is 24 are 3 and 5.
a = first integer
a = 2 k + 1
b = second integer
Becouse odd integers consecutive are consecutive:
b = a + 2 = 2 k + 1 + 2 = 2 k + 3
k is some integer
The square of the first, added to 3 times the second is 24 mean:
a² + 3 b = 24
( 2 k + 1 )² + 3 ∙ ( 2 k + 3 ) = 24
( 2 k )² + 2 ∙ 2 k ∙ 1 + 1² + 3 ∙ 2 k + 3 ∙ 3 = 24
4 k² + 4 k + 1 + 6 k + 9 = 24
4 k² + 10 k + 10 = 24
4 k² + 10 k + 10 - 24 = 0
4 k² + 10 k - 14 = 0
The solutions are :
k = - 7 / 2 and k = 1
k is some integer so k = 1
Yours two consecutive odd integers :
a = 2 k + 1 = 2 ∙ 1 + 1 = 2 + 1 = 3
b = 2 k + 3 = 2 ∙ 1 + 3 = 2 + 1 = 5
Proof:
3² + 3 ∙ 5 = 9 + 15 = 24