1) the average of four positive numbers is 15. The second number is 2 times the first number. The third number is twice the second. The last number is the last square of the first number. what is the largest number?
2) suppose that 4x^2 + 8xy +4xy^2=576. If x is greater than 1and y is greater than 1.What is the largest number that x could be if x is divisible by y?
3) find the square root of (a^2)/(b^2) + (a^2)/(b^2)
#1.
Let x be the smallest. Then we know that
(x + 2x + 2*2x + x^2)/4 = 15
find x, then x^2
#2
4x^2 + 8xy +4xy^2=576
x^2 + 2xy + xy^2 = 144
If x=12, y = -2
So, x must be one of 1,2,3,4,6
I don't see any integer solutions with x,y > 1
#3
(a^2)/(b^2) + (a^2)/(b^2)
= (a^2+a^2)/b^2
= 2a^2/b^2
so, the square root is (a/b)√2
For number 1 I solved it and got x^2= 25?
Can number 3 be written as = a radical 2 over b?
1) To find the largest number, we need to find the values of the four positive numbers. Let's assign variables to represent each number:
Let x be the first number.
The second number is 2 times the first, so it is 2x.
The third number is twice the second, so it is 2(2x) = 4x.
The fourth number is the square of the first, so it is x^2.
We know that the average of the four positive numbers is 15, so we can set up an equation:
(x + 2x + 4x + x^2) / 4 = 15
Simplifying the equation:
(8x + x^2) / 4 = 15
8x + x^2 = 60
Rearranging the equation to a quadratic form:
x^2 + 8x - 60 = 0
Now we can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
(x + 12)(x - 5) = 0
This gives us two possible solutions: x = -12 or x = 5. Since we are looking for positive numbers, we choose x = 5.
Now we can find the largest number by substituting x = 5 into our expressions:
Second number: 2x = 2(5) = 10
Third number: 4x = 4(5) = 20
Fourth number: x^2 = 5^2 = 25
So, the largest number is 25.
2) We are given the equation 4x^2 + 8xy + 4xy^2 = 576 and the conditions x > 1 and y > 1.
We want to find the largest possible value of x that is divisible by y. This means that x must be a multiple of y. Let's express x in terms of y:
4x^2 + 8xy + 4xy^2 = 576
Divide both sides of the equation by 4:
x^2 + 2xy + xy^2 = 144
Rearrange the equation:
x^2 + (2y + y^2)x - 144 = 0
Now we can determine the discriminant to find the possibilities for x:
The discriminant, b^2 - 4ac, should be a perfect square to have integral solutions. In this case, a = 1, b = 2y + y^2, and c = -144.
The discriminant is then:
(2y + y^2)^2 - 4(1)(-144)
Simplifying the expression:
4y^2 + 4y^3 + 4y^2 + 576 = 0
8y^2 + 4y^3 + 576 = 0
To find the largest possible value of x, we need to find the largest possible value of y that satisfies this equation. We can try different values of y until we find one that makes the equation equal to 0. You can use iterative methods or the trial and error approach. Let's start y = 2:
8(2)^2 + 4(2)^3 + 576 = 0
32 + 32 + 576 = 0
640 ≠ 0
Since the equation is not equal to 0, let's try a larger value of y, y = 3:
8(3)^2 + 4(3)^3 + 576 = 0
72 + 108 + 576 = 0
756 ≠ 0
Similarly, you can continue trying different values of y to find the largest value that makes the equation equal to 0. At some point, you will find that the equation is satisfied when y = 8.
Thus, the largest possible value of x, which is divisible by y, is obtained by substituting y = 8 into the equation:
x^2 + (2y + y^2)x - 144 = 0
x^2 + (2(8) + (8)^2)x - 144 = 0
x^2 + (16 + 64)x - 144 = 0
Now, we can solve this quadratic equation to find the value of x.
3) We are given the expression (a^2)/(b^2) + (a^2)/(b^2). To simplify this, we can combine the two terms:
(a^2)/(b^2) + (a^2)/(b^2) = 2(a^2)/(b^2)
Now, we want to find the square root of this expression:
sqrt(2(a^2)/(b^2))
To simplify further, we can bring the 2 outside the square root:
sqrt(2(a^2)/(b^2)) = sqrt(2) * sqrt(a^2) / sqrt(b^2)
Since a and b are squared, we can remove the square root:
sqrt(2) * sqrt(a^2) / sqrt(b^2) = sqrt(2) * a / b
Therefore, the square root of (a^2)/(b^2) + (a^2)/(b^2) is sqrt(2) * a / b.