Question

If the sum of two numbers is 10 and the difference of their squares is 20, then these numbers are:

• A. 8 and 2
• B. 6 and 4
• C. 7 and 3
• D. 5 and 5

Hint:

If both \(x+y\) and \(x^2-y^2\) are given, first use \[ x^2-y^2=(x-y)(x+y) \] to find \(x-y\), then solve the two linear equations.

Answer 1

The Correct Option is B

Step 1: Form the equations. Let the numbers be \(x\) and \(y\). Given: \[ x+y=10 \] and \[ x^2-y^2=20 \] Using the identity: \[ (x-y)(x+y)=20 \] Substituting \(x+y=10\), \[ 10(x-y)=20 \] \[ x-y=2 \]

Step 2: Solve simultaneously. \[ x+y=10 \] \[ x-y=2 \] Adding, \[ 2x=12 \] \[ x=6 \] Substituting into \(x+y=10\), \[ y=4 \] Thus, the two numbers are \[ {6 \text{ and } 4} \] Hence, the correct answer is Option (B).