If 2ab + 5cd = 5 and abcd = 1, what is the value of 4a^2b^2+ 25 c^2d^2
We know:
(2ab + 5cd)^2 = 4a^2 + 25c^2d^2 + 20abcd
plug in the values given, and finish it
To find the value of 4a^2b^2 + 25c^2d^2, we can first simplify the expression 4a^2b^2 and 25c^2d^2 separately before adding them together.
Starting with 4a^2b^2, we can rewrite it as 2^2 * (ab)^2. Since the given equation is 2ab + 5cd = 5, we can use this equation to substitute the value of 2ab in terms of (5 - 5cd):
4a^2b^2 = 4(5 - 5cd)^2
Now, let's simplify 25c^2d^2. We can rewrite it as 5^2 * (cd)^2. Again, referring to the given equation 2ab + 5cd = 5, we can express 5cd as (5 - 2ab):
25c^2d^2 = 25(5 - 2ab)^2
So, the expression 4a^2b^2 + 25c^2d^2 can be written as:
4(5 - 5cd)^2 + 25(5 - 2ab)^2
Now, let's substitute the value of abcd = 1 into the equation 2ab + 5cd = 5. Solving for ab in terms of cd, we get:
2ab = 5 - 5cd
ab = (5 - 5cd) / 2
Substituting this value into the expression, we have:
4(5 - 5cd)^2 + 25(5 - 2 * (5 - 5cd) / 2)^2
Simplifying further:
4(5 - 5cd)^2 + 25(5 - 2 * (5 - 5cd) / 2)^2
= 4(5 - 5cd)^2 + 25(5 - (5 - 5cd))^2
= 4(5 - 5cd)^2 + 25(5cd)^2
= 4(25 - 50cd + 25cd^2) + 25(25c^2d^2)
= 100 - 200cd + 100cd^2 + 625c^2d^2
Since we know that abcd = 1, we can also substitute this value into the equation:
abcd = 1
ab * cd = 1
((5 - 5cd) / 2) * cd = 1
(5cd - 5cd^2) / 2 = 1
5cd - 5cd^2 = 2
5 - 5cd = 2 / cd
Substituting this back into the expression, we have:
100 - 200cd + 100cd^2 + 625c^2d^2
= 100 - 200cd + 100cd^2 + 625c^2d^2
= 100 - 200(2 / cd) + 100(2 / cd)^2 + 625c^2d^2
= 100 - 400 / cd + 400 / cd^2 + 625c^2d^2
Thus, the value of 4a^2b^2 + 25c^2d^2 is 100 - 400 / cd + 400 / cd^2 + 625c^2d^2.
To find the value of 4a^2b^2 + 25c^2d^2, we can use the given equations and solve for the values of a, b, c, and d. Let's go step by step:
1. Start with the equation 2ab + 5cd = 5.
2. Rearrange this equation to solve for a: 2ab = 5 - 5cd.
3. Divide both sides by 2b to isolate a: a = (5 - 5cd) / (2b).
4. Next, substitute the value of a into the equation abcd = 1:
(5 - 5cd) / (2b) * b * cd = 1.
5. Simplify the equation: (5 - 5cd) * cd = 2b.
6. Distribute the cd: 5cd - 5c^2d^2 = 2b.
7. Rearrange the equation to solve for b: 2b = 5cd - 5c^2d^2.
8. Divide both sides by 2: b = (5cd - 5c^2d^2) / 2.
Now, we have expressions for a and b in terms of c and d. Let's substitute these values into the expression 4a^2b^2 + 25c^2d^2:
4a^2b^2 + 25c^2d^2 = 4[(5 - 5cd) / (2b)]^2 + 25c^2d^2.
9. Substitute the value of b: 4[(5 - 5cd) / (2[(5cd - 5c^2d^2) / 2])]^2 + 25c^2d^2.
10. Simplify the expression:
4[(5 - 5cd) / (5cd - 5c^2d^2)]^2 + 25c^2d^2.
You can now further simplify the expression and calculate the final value using the given values of c and d.