Question

M2+n2=9 (m-n)^2=5 mn-?

Answer 1

(m-n)²= m²-2mn+n²=5

9-2mn=5

4=2mn

mn=2

Answer 2

To solve the given equations, let's start by expanding the second equation:

(m - n)^2 = m^2 - 2mn + n^2 = 5

Now, we can rearrange the first equation to express either m or n in terms of the other variable. Let's solve it for m:

m^2 + n^2 = 9

m^2 = 9 - n^2

m = √(9 - n^2)

Now, substitute this value of m into the second equation:

(√(9 - n^2) - n)^2 = 5

Expanding the equation:

(9 - n^2) - 2n√(9 - n^2) + n^2 = 5

Simplifying:

2n√(9 - n^2) = 4

Divide both sides by 2n:

√(9 - n^2) = 2

Square both sides to eliminate the square root:

9 - n^2 = 4

Rearrange the equation:

n^2 = 9 - 4

n^2 = 5

Taking the square root of both sides:

n = ±√5

Now, substitute the value of n back into the first equation to find m:

m^2 + (±√5)^2 = 9

m^2 + 5 = 9

m^2 = 9 - 5

m^2 = 4

m = ±2

Therefore, we have two sets of solutions:

1. When n = √5, m = 2
2. When n = -√5, m = -2

Now, to find the value of mn, we can substitute these values into either of the original equations:

For the first set of solutions (n = √5, m = 2):

mn = (√5)(2) = 2√5

For the second set of solutions (n = -√5, m = -2):

mn = (-√5)(-2) = 2√5

So, mn = 2√5 for both sets of solutions.