Two numbers differ by 6. The sun of their squares is 146. Find the number.
Let the two numbers be x and y.
We know that the two numbers differ by 6, so we can write the equation: x - y = 6. (1)
We also know that the sum of their squares is 146, so we can write the equation: x^2 + y^2 = 146. (2)
To solve this system of equations, we can use substitution.
From equation (1), we can express x in terms of y: x = y + 6.
Substituting this expression for x in equation (2), we have: (y + 6)^2 + y^2 = 146.
Expanding and simplifying: y^2 + 12y + 36 + y^2 = 146.
Combining like terms: 2y^2 + 12y + 36 = 146.
Rearranging the equation: 2y^2 + 12y + 36 - 146 = 0.
Simplifying: 2y^2 + 12y - 110 = 0.
Dividing through the equation by 2: y^2 + 6y - 55 = 0.
Factoring: (y + 11)(y - 5) = 0.
Setting each factor equal to zero:
y + 11 = 0,
y = -11.
y - 5 = 0,
y = 5.
So we have two possible solutions for y: y = -11 or y = 5.
If y = -11, substituting this value in the equation x = y + 6 gives us:
x = -11 + 6,
x = -5.
If y = 5, substituting this value in the equation x = y + 6 gives us:
x = 5 + 6,
x = 11.
Therefore, the two numbers are -5 and -11 or 11 and 5.
Let's solve this step by step.
Step 1: Let's call the two numbers x and y.
We know that y is 6 more than x, so we can write it as:
y = x + 6.
Step 2: The sum of their squares is 146, so we can write it as an equation:
x^2 + y^2 = 146.
Step 3: We substitute the value of y from Step 1 into the equation from Step 2:
x^2 + (x + 6)^2 = 146.
Step 4: Expand the equation:
x^2 + (x^2 + 12x + 36) = 146.
Step 5: Combine like terms:
2x^2 + 12x + 36 = 146.
Step 6: Move all terms to one side of the equation to set it equal to zero:
2x^2 + 12x + 36 - 146 = 0.
Step 7: Simplify:
2x^2 + 12x - 110 = 0.
Step 8: Divide the equation by 2 to simplify:
x^2 + 6x - 55 = 0.
Step 9: Factor the quadratic equation:
(x + 11)(x - 5) = 0.
Step 10: Set each factor equal to zero and solve for x:
x + 11 = 0 or x - 5 = 0.
If x + 11 = 0, then x = -11.
If x - 5 = 0, then x = 5.
Step 11: So, the two numbers are either -11 and -5, or 5 and 11.
Therefore, the number is either -11 and -5, or 5 and 11.
To solve this problem, let's assign variables to the numbers. Let's call the first number "x" and the second number "y".
We are given that the two numbers differ by 6, so we can write the following equation:
x - y = 6 (Equation 1)
We are also given that the sum of their squares is 146, which gives us the following equation:
x² + y² = 146 (Equation 2)
Now we have a system of two equations with two variables. To solve this system, we can use the method of substitution or elimination.
Let's solve it using the substitution method.
From Equation 1, we can isolate x:
x = y + 6
Now we substitute this expression for x in Equation 2:
(y + 6)² + y² = 146
Expanding the square:
y² + 12y + 36 + y² = 146
Combining like terms:
2y² + 12y + 36 = 146
Rearranging the equation:
2y² + 12y + 36 - 146 = 0
2y² + 12y - 110 = 0
Dividing the equation by 2 to simplify:
y² + 6y - 55 = 0
Now we can solve this quadratic equation for y by factoring or using the quadratic formula. After finding the values of y, we can substitute them back into Equation 1 to find the corresponding values of x.
Using the quadratic formula:
y = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 1, b = 6, and c = -55.
Plugging in the values:
y = (-6 ± √(6² - 4(1)(-55))) / 2(1)
Simplifying:
y = (-6 ± √(36 + 220)) / 2
y = (-6 ± √256) / 2
y = (-6 ± 16) / 2
Now we have two possible values for y:
y₁ = (-6 + 16) / 2 = 5
y₂ = (-6 - 16) / 2 = -11
We substitute these values back into Equation 1 to find the corresponding values of x:
For y = 5:
x = y + 6 = 5 + 6 = 11
For y = -11:
x = y + 6 = -11 + 6 = -5
So the two numbers can be either 11 and 5, or -5 and -11.