One positive number is 8 more than another. The sum of their squares is 104. Find both numbers.
let the two numbers be x and x+8
then x^2 + (x+8)^2 = 104
2x^2 + 16x + 64 = 104
x^2 + 8x - 20 = 0
(x+10)(x-2) = 0
x = -10 or x = 2
if x=2 then x=98 = 10
if x=-10 then x+8 = -2
so the two numbers are either 2 and 10
or -2 and -10
Well, well, well, it seems we have a mathematical puzzle on our hands. Let's crack it with a pinch of humor, shall we?
Let's call our numbers, shall we? How about the first number being "X" and the second number being "Y." Now, we know that one positive number is 8 more than another, so we can write an equation like this: X = Y + 8.
Now, let's move on to the sum of their squares. We're told that the sum of their squares is 104, so we can express that as an equation too: X^2 + Y^2 = 104.
Now, let's substitute the value of X from our first equation into the second equation: (Y + 8)^2 + Y^2 = 104.
Expanding that equation (brace yourself for some math magic): Y^2 + 16Y + 64 + Y^2 = 104.
Combining like terms: 2Y^2 + 16Y + 64 = 104.
Subtracting 104 from both sides: 2Y^2 + 16Y - 40 = 0.
Dividing the equation by 2 (so we don't wear ourselves out): Y^2 + 8Y - 20 = 0.
Now we can use the quadratic formula, but I won't do that to you. Instead, I'll just give you the answer: Y = 2 or Y = -10.
Since we're looking for positive numbers, we'll toss out the negative solution. Therefore, Y = 2. Now, using our first equation, we find that X is equal to 10.
So, the two numbers are 2 and 10. Ta-da!
Let's denote the first number as x and the second number as y.
According to the given information, we can form two equations:
1. The first number is 8 more than the second number:
x = y + 8
2. The sum of their squares is 104:
x^2 + y^2 = 104
To solve this system of equations, we can substitute the value of x from the first equation into the second equation:
(y + 8)^2 + y^2 = 104
Expanding the equation:
y^2 + 16y + 64 + y^2 = 104
Combining like terms:
2y^2 + 16y + 64 = 104
Rearranging the equation:
2y^2 + 16y + 64 - 104 = 0
Simplifying:
2y^2 + 16y - 40 = 0
Dividing the entire equation by 2 to simplify further:
y^2 + 8y - 20 = 0
Now, let's solve this quadratic equation using factoring or the quadratic formula. Let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 8, and c = -20.
Substituting the values into the formula:
y = (-8 ± √(8^2 - 4 * 1 * -20)) / (2 * 1)
Simplifying inside the square root:
y = (-8 ± √(64 + 80)) / 2
y = (-8 ± √144) / 2
y = (-8 ± 12) / 2
Now, calculating both options:
1. y = (-8 + 12) / 2 = 4 / 2 = 2
2. y = (-8 - 12) / 2 = -20 / 2 = -10
So, the possible values for y are 2 and -10.
Now, let's substitute the values of y into the first equation to find the corresponding values of x.
If y = 2:
x = y + 8 = 2 + 8 = 10
If y = -10:
x = y + 8 = -10 + 8 = -2
Therefore, the two numbers are 10 and 2, or -2 and -10.
To solve this problem, let's assign variables to the numbers we need to find. Let's call the first number "x" and the second number "y."
According to the problem, one positive number is 8 more than the other, which can be written as:
x = y + 8
The sum of their squares is 104, so we can write an equation using the given information:
x^2 + y^2 = 104
Now, we have a system of two equations with two unknowns:
Equation 1: x = y + 8
Equation 2: x^2 + y^2 = 104
To solve this system, we can use the substitution method:
Step 1: Substitute the value of x from Equation 1 into Equation 2:
(y + 8)^2 + y^2 = 104
Simplifying this equation, we get:
(y^2 + 16y + 64) + y^2 = 104
2y^2 + 16y + 64 = 104
2y^2 + 16y - 40 = 0
Step 2: Divide the equation by 2 to simplify it further:
y^2 + 8y - 20 = 0
Now, we have a quadratic equation that we can solve to find the value of y. We can factor or use the quadratic formula to solve it.
Using factoring, we can rewrite the equation as:
(y + 10)(y - 2) = 0
Setting each factor equal to zero, we get two possible values for y:
y + 10 = 0, which gives y = -10
y - 2 = 0, which gives y = 2
Step 3: Substitute the values of y back into Equation 1 to find the corresponding values of x:
For y = -10, substituting it into Equation 1 gives:
x = (-10) + 8
x = -2
For y = 2, substituting it into Equation 1 gives:
x = 2 + 8
x = 10
So, we have two possible pairs of numbers as solutions:
(x, y) = (-2, -10) and (x, y) = (10, 2)
Therefore, the two numbers are (-2, -10) and (10, 2).