Finding Two Numbers

Video Explanation

Question

The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.

Solution

Step 1: Let the Variables

Let the first number = \(x\)

Let the second number = \(y\)

Step 2: Form the First Equation

\[ x + y = 1000 \quad (1) \]

Step 3: Use Identity for Difference of Squares

\[ x^2 – y^2 = 256000 \]

Using identity:

\[ x^2 – y^2 = (x – y)(x + y) \]

So,

\[ (x – y)(x + y) = 256000 \]

Substitute equation (1):

\[ (x – y)(1000) = 256000 \]

\[ x – y = 256 \quad (2) \]

Step 4: Solve the Equations

We now have:

\[ x + y = 1000 \]

\[ x – y = 256 \]

Add both equations:

\[ 2x = 1256 \]

\[ x = 628 \]

Step 5: Find the Value of y

\[ y = 1000 – 628 \]

\[ y = 372 \]

Conclusion

First number:

\[ \boxed{628} \]

Second number:

\[ \boxed{372} \]

Final Answer (For Exam)

The two numbers are 628 and 372.