The sum of squares of two numbers is 520 and there difference is 4 what is smaller number.
let the smaller be x
then the larger is x+4
x^2 + (x+4)^2 = 520
x^2 + x^2 + 8x + 16 = 520
2x^2 + 8x - 504
x^2 + 4x - 252 = 0
(x+18)(x-14) = 0
x = -18 or x = 14
2 cases:
if x = -18 , the smaller is -18, the larger is -14
if x = 14, then the larger is 18
Let's assume the two numbers as x and y. We can begin by setting up the given information as equations:
Equation 1: x^2 + y^2 = 520 (sum of squares is 520)
Equation 2: x - y = 4 (difference is 4)
Now we can solve these equations step-by-step to find the value of the smaller number.
Step 1: Rearrange Equation 2 to solve for x.
x = y + 4
Step 2: Substitute the value of x from Step 1 into Equation 1.
(y + 4)^2 + y^2 = 520
Step 3: Expand and simplify the equation from Step 2.
y^2 + 8y + 16 + y^2 = 520
2y^2 + 8y + 16 = 520
Step 4: Subtract 520 from both sides of the equation.
2y^2 + 8y - 504 = 0
Step 5: Divide the equation from Step 4 by 2 to simplify.
y^2 + 4y - 252 = 0
Step 6: Factorize the equation from Step 5 or use the quadratic formula to find the values of y.
(y - 12)(y + 21) = 0
Step 7: Solve for y by setting each factor equal to zero.
y - 12 = 0 --> y = 12
y + 21 = 0 --> y = -21
Step 8: Since we are looking for the smaller number, we choose the value of y to be 12.
Therefore, the smaller number is 12.
To solve this problem, we need to set up a system of equations.
Let's assume the smaller number is 'x' and the larger number is 'y'.
According to the problem, the sum of squares of two numbers is 520, so we can write the equation:
x^2 + y^2 = 520 ----(equation 1)
It is also given that their difference is 4, so we can write the equation:
y - x = 4 ----(equation 2)
Now, we have a system of two equations with two variables. We can use substitution or elimination method to solve for 'x' and 'y'.
Let's solve equation 2 for 'y':
y = 4 + x
Now, substitute this value of 'y' into equation 1:
x^2 + (4 + x)^2 = 520
Expand and simplify the equation:
x^2 + 16 + 8x + x^2 = 520
Combine like terms:
2x^2 + 8x + 16 = 520
Rearrange the equation:
2x^2 + 8x - 504 = 0
Now, we can solve this quadratic equation for 'x' using factoring, completing the square, or the quadratic formula. Factoring or completing the square methods may be a bit complicated, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 8, and c = -504. Plugging in these values:
x = (-8 ± √(8^2 - 4 * 2 * -504)) / (2 * 2)
Simplify the expression inside the square root:
x = (-8 ± √(64 + 4032)) / 4
x = (-8 ± √(4096)) / 4
x = (-8 ± 64) / 4
Now, we have two possible values for 'x':
1) x = (-8 + 64) / 4 = 56 / 4 = 14
2) x = (-8 - 64) / 4 = -72 / 4 = -18
These are the potential values for the smaller number.
However, the problem specifically asks for the smaller number, so we choose the smaller value, which is -18.
Therefore, the smaller number is -18.