Real numbers x and y have a difference of 16 and a product of 23. Find x^2+y^2.
larger --- x
smaller --- y
x-y = 16
xy = 23
x^2 + y^2 = (x - y)^2 + 2xy
= 16^2 + 46 = 302
x^2+y^2=(x-y)^2+2xy=16^2+2*23=256+46=302
Well, this problem seems a bit of a mathematical circus! Let's call our real numbers x and y. We know that their difference is 16, so we can write an equation: x - y = 16.
Next, we're told that their product is 23. Now, that's a tricky one! Let's write another equation: xy = 23.
Now, how can we use these two equations to find the value of x^2 + y^2? This problem needs a little clown magic!
Let's start by squaring the equation x - y = 16. We get (x - y)^2 = 256. Now, let's expand that: x^2 - 2xy + y^2 = 256.
Hmm, this looks promising! We know that xy = 23, so let's substitute that in: x^2 - 2(23) + y^2 = 256. Simplify further: x^2 + y^2 = 256 + 46.
Finally, after a little somersaulting, we get x^2 + y^2 = 302! Voila! So, the answer is 302.
To find the value of x and y, we can form a system of equations based on the given information.
Let's assume x and y are the two real numbers we're looking for.
According to the given information, we have two equations:
1) x - y = 16 (the difference between x and y is 16)
2) x * y = 23 (the product of x and y is 23)
To solve this system of equations, we can use substitution or elimination.
Using the substitution method:
From equation 1, we can rewrite it as x = y + 16.
Now substitute this expression for x in equation 2:
(y + 16) * y = 23
Expanding this equation:
y^2 + 16y = 23
Rearranging the equation:
y^2 + 16y - 23 = 0
To solve this quadratic equation, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 1, b = 16, and c = -23.
Substituting these values into the quadratic formula, we have:
y = (-16 ± √(16^2 - 4 * 1 * -23)) / (2 * 1)
Simplifying further:
y = (-16 ± √(256 + 92)) / 2
y = (-16 ± √348) / 2
y = (-16 ± √(4 * 87)) / 2
y = (-16 ± 2√87) / 2
y = -8 ± √87
So, we have two possible solutions for y:
1) y = -8 + √87
2) y = -8 - √87
To find the corresponding values of x, we can substitute these y values back into the equation x = y + 16:
For y = -8 + √87:
x = (-8 + √87) + 16
x = 8 + √87
For y = -8 - √87:
x = (-8 - √87) + 16
x = 8 - √87
Therefore, the two possible solutions for x and y are:
1) x = 8 + √87, y = -8 + √87
2) x = 8 - √87, y = -8 - √87
To find x^2 + y^2, we can substitute these values into the expression:
1) (8 + √87)^2 + (-8 + √87)^2
2) (8 - √87)^2 + (-8 - √87)^2
Now we can compute these expressions to find the final answer.