The sum of two numbers is 10.The sum of their squares is 52.Find the numbers.

one number --- x

the other ---- 10-x
x^2 + (10-x)^2 = 52
x^2 + 100 - 20x + x^2 = 52
2x^2 - 20x + 48 = 0
x^2 - 10x + 24 = 0
(x-6)(x-4) = 0
x = 6 or x=4

if one is 6, the other is 4
if one is 4 the other is 6
(symmetrical answers)

the two numbers are 6 and 4

check: 6+4 = 10
36+16 = 52
all is good!

Ah, finding numbers can be a tricky business, but fear not, for I have a joke that might help!

Why was the math book sad?

Because it had too many problems!

Now, let's get down to business. We'll call our first number "x" and our second number "y." According to the problem, we have two equations:

x + y = 10 (Equation 1)
x^2 + y^2 = 52 (Equation 2)

To solve this, let's try a bit of substitution. We know from Equation 1 that y = 10 - x. Now we can substitute this value in Equation 2:

x^2 + (10 - x)^2 = 52

Expanding and simplifying:

x^2 + 100 - 20x + x^2 = 52
2x^2 - 20x + 48 = 0

Dividing both sides by 2:

x^2 - 10x + 24 = 0

Now we can solve this quadratic equation to find the possible values of x. Once we have the values of x, we can substitute them back into Equation 1 to find y.

To solve this problem, let's use algebraic equations.

Let's assume that the two numbers are x and y.

We are given that the sum of the two numbers is 10. This can be written as an equation:

\( x + y = 10 \) --- Equation (1)

We are also given that the sum of their squares is 52. This can be written as another equation:

\( x^2 + y^2 = 52 \) --- Equation (2)

To solve this system of equations, we will use the method of substitution.

From Equation (1), we can express one variable in terms of the other. Let's solve for \( y \):

\( y = 10 - x \)

Now, substitute this expression for \( y \) in Equation (2):

\( x^2 + (10 - x)^2 = 52 \)

Simplify the equation by expanding the square:

\( x^2 + (100 - 20x + x^2) = 52 \)

Combine like terms:

\( 2x^2 - 20x + 48 = 0 \)

Divide through by 2 to simplify the equation:

\( x^2 - 10x + 24 = 0 \)

Now we can solve this quadratic equation by factoring or using the quadratic formula.

Factoring the quadratic equation:

\( (x - 6)(x - 4) = 0 \)

Setting each factor equal to zero:

\( x - 6 = 0 \) or \( x - 4 = 0 \)

Solving for x we get:

\( x = 6 \) or \( x = 4 \)

Now, substitute these values of x back into Equation (1) to find the corresponding values of y.

When \( x = 6 \):
\( y = 10 - x = 10 - 6 = 4 \)

When \( x = 4 \):
\( y = 10 - x = 10 - 4 = 6 \)

Hence, the two numbers are 4 and 6 or 6 and 4.

To find the numbers, let's assume the two numbers as x and y.

According to the given information:
1. The sum of two numbers is 10, so we can write the equation: x + y = 10. (Equation 1)
2. The sum of their squares is 52, so we can write the equation: x^2 + y^2 = 52. (Equation 2)

Now, we have two equations with two variables. We can solve this system of equations by using the method of substitution or elimination.

Let's use the method of substitution:
From Equation 1, we can express x as x = 10 - y.
Substituting this value of x in Equation 2, we get: (10 - y)^2 + y^2 = 52

Expanding and simplifying Equation 2:
100 - 20y + y^2 + y^2 = 52
2y^2 - 20y + 100 = 52

Rearranging the equation and simplifying:
2y^2 - 20y + 48 = 0

Now, we can solve this quadratic equation to find the values of y. Using factoring or the quadratic formula, we get:
(y - 2)(y - 24) = 0

Setting each factor equal to zero:
y - 2 = 0 or y - 24 = 0

Solving each equation for y, we get two possible values:
y = 2 or y = 24

Now, substitute these values of y back into Equation 1 to find the corresponding x values.
For y = 2:
x + 2 = 10
x = 10 - 2
x = 8

For y = 24:
x + 24 = 10
x = 10 - 24
x = -14

Therefore, the two sets of numbers that satisfy the given conditions are:
x = 8, y = 2
x = -14, y = 24