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The sum of two numbers is 25, and the sum of their squares is 313.

Find the numbers. Express the other number in terms of [tex]$x$[/tex].

Answer :

To solve the problem where the sum of two numbers is 25 and the sum of their squares is 313, let's go through the steps to find the numbers.

1. Define variables:
- Let the two numbers be [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

2. Set up the equations:
- From the problem, we know:
[tex]\[
x + y = 25
\][/tex]
- We also know the sum of their squares:
[tex]\[
x^2 + y^2 = 313
\][/tex]

3. Express one variable in terms of the other:
- From the first equation, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 25 - x
\][/tex]

4. Substitute into the second equation:
- Substitute [tex]\( y = 25 - x \)[/tex] into the equation for the sum of squares:
[tex]\[
x^2 + (25 - x)^2 = 313
\][/tex]

5. Simplify the quadratic equation:
- Expand [tex]\( (25 - x)^2 \)[/tex]:
[tex]\[
(25 - x)^2 = 625 - 50x + x^2
\][/tex]
- Substitute back into the equation:
[tex]\[
x^2 + 625 - 50x + x^2 = 313
\][/tex]
- Combine like terms:
[tex]\[
2x^2 - 50x + 625 = 313
\][/tex]
- Simplify the equation by subtracting 313 from both sides:
[tex]\[
2x^2 - 50x + 312 = 0
\][/tex]

6. Solve the quadratic equation:
- To solve [tex]\( 2x^2 - 50x + 312 = 0 \)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- For our equation [tex]\( 2x^2 - 50x + 312 \)[/tex], [tex]\( a = 2 \)[/tex], [tex]\( b = -50 \)[/tex], [tex]\( c = 312 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-50)^2 - 4(2)(312)
\][/tex]
[tex]\[
2500 - 2496 = 4
\][/tex]
- Find the solutions:
[tex]\[
x = \frac{50 \pm \sqrt{4}}{4}
\][/tex]
[tex]\[
x = \frac{50 \pm 2}{4}
\][/tex]
[tex]\[
x_1 = \frac{52}{4} = 13
\][/tex]
[tex]\[
x_2 = \frac{48}{4} = 12
\][/tex]

7. Find the corresponding [tex]\( y \)[/tex] values:
- If [tex]\( x = 13 \)[/tex], then [tex]\( y = 25 - 13 = 12 \)[/tex].
- If [tex]\( x = 12 \)[/tex], then [tex]\( y = 25 - 12 = 13 \)[/tex].

Thus, the two numbers are 13 and 12.

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Rewritten by : Brahmana