x^2+y^2=86 and xy=(-16), Find the value of
(x-y)^2
Please help i am a bit tense today so i can't figure it out.
involves a bit of mathematical "trickery"
we know
(x-y)^2 = x^2 - 2xy + y^2
= x^2 + y^2 - 2xy
= 86 + 32 = 118
real easy, right ?
To find the value of (x-y)^2, we can use the given equations and solve step-by-step. Let's start:
1. Given x^2 + y^2 = 86. We can rewrite this equation as x^2 = 86 - y^2.
2. Now substitute xy = -16 into equation 1. We'll get x(86-y^2) = -16.
3. Distribute x into the equation: 86x - xy^2 = -16.
4. Replace xy with -16: 86x - (-16)y^2 = -16. Simplify further to: 86x + 16y^2 = -16.
5. Now we will solve these two equations simultaneously. Substitute x from equation 2 into equation 5:
86(86-y^2) + 16y^2 = -16.
Simplifying further: 7396 - 86y^2 + 16y^2 = -16.
Combine like terms: -70y^2 = -7412.
Divide both sides by -70 to isolate y^2: y^2 = -7412 / -70.
y^2 = 105.88.
6. Taking the square root of both sides to solve for y: y = ±√105.88.
7. Now substitute the value of y into equation 2: x(±√105.88) = -16.
8. Solve for x by dividing both sides by ±√105.88: x = -16 / (√105.88).
Hence, we have two solutions: (x, y) = (-16 / √105.88, √105.88) and (x, y) = (-16 / √105.88, -√105.88).
9. Finally, to find the value of (x-y)^2, substitute the values of x and y into the expression (x-y)^2 and simplify:
For the first solution (-16 / √105.88, √105.88):
(x-y)^2 = (-16 / √105.88 - √105.88)^2.
Simplify further using the values of x and y.
For the second solution (-16 / √105.88, -√105.88):
(x-y)^2 = (-16 / √105.88 + √105.88)^2.
Simplify further using the values of x and y.
By following these steps, you can find the value of (x-y)^2 in this specific problem.