If P^2+q^2 = 25 and 2 pq=10. What is the value of (p-q) ^2?
Well, let's put our thinking hats on and solve this math puzzle!
Given that P^2 + q^2 = 25 and 2pq = 10, we can manipulate these equations to find the value of (p-q)^2.
First, let's square the second equation, which gives us (2pq)^2 = 100. Expanding this gives us 4p^2q^2 = 100.
Now let's manipulate the first equation by subtracting 2pq from both sides, which gives us P^2 - 2pq + q^2 = 25 - 10. Simplifying this equation gives us P^2 - 2pq + q^2 = 15.
Now, let's use these two equations to solve for (p-q)^2. Rearranging the equation P^2 - 2pq + q^2 = 15, we get (p-q)^2 = 15 - 4p^2q^2.
Substituting the value of 4p^2q^2 from our earlier calculation, we get (p-q)^2 = 15 - 100.
And finally, simplifying further, (p-q)^2 = -85.
Ta-da! The value of (p-q)^2 is -85!
Just remember, in the world of math, sometimes the answers can be a little silly.
To solve this problem, we can use the fact that (a-b)^2 = a^2 - 2ab + b^2.
Given that P^2 + q^2 = 25, we know that (P + q)(P - q) = P^2 - q^2.
Substituting the given values, we have (P + q)(P - q) = 25 - 2pq.
Since 2pq = 10, we can substitute it into the equation:
(P + q)(P - q) = 25 - 2(10)
(P + q)(P - q) = 25 - 20
(P + q)(P - q) = 5
So, the value of (P - q)^2 is 5.
To find the value of (p - q)², we need to simplify the expression (p - q)² using the given equations.
Let's start by expanding (p - q)²:
(p - q)² = (p - q)(p - q)
= p(p - q) - q(p - q)
= p² - pq - qp + q²
= p² - 2pq + q²
Now, we can substitute the values we have:
p² + q² = 25 (Equation 1)
2pq = 10 (Equation 2)
Using Equation 2, we can rewrite it as:
pq = 5
Now, let's substitute the value of pq in the expression for (p - q)²:
(p - q)² = p² - 2pq + q²
= p² - 2(5) + q² (since pq = 5)
= p² - 10 + q²
Since we know that p² + q² = 25 (Equation 1), we can substitute it in the expression:
(p - q)² = (p² + q²) - 10
Substituting the known value, we have:
(p - q)² = 25 - 10
= 15
Therefore, the value of (p - q)² is 15.
note that
(p-q)^2 = p^2 - 2pq + q^2