A square-of-a-difference identity helps when a system gives a sum of squares and a difference but asks for a product. Square the difference and expand it to bring the product into an equation with the known sum. You don't need to find either coordinate. Watch the negative middle term: changing its sign changes the product.
Hints
- Hint 1
A system means the same and satisfy both equations. You need their product, not their individual values. What happens if you square the given difference?
- Hint 2
The square of a difference is . Its middle term is negative because multiplying by gives twice. Which part of this expansion does the other equation give you?
Step-by-step
Square the difference to find the product
Step 1Square the given difference
No Desmos needed. An identity gives the requested product without solving for and separately. Square both sides of ; is :
- Step 2
Expand to reveal the product
Write the square of a difference as multiplication:
Multiply each term. Both cross terms are negative:
Combine the two terms:
The squared difference reveals the product without finding either coordinate.
- Step 3
Use the given sum of squares
The first equation gives as one known sum. Group those terms in the expansion:
Replace the sum of squares with :
- Step 4
Isolate the requested product
Subtract from both sides:
Divide by :
The value of is , even if the system has more than one real solution. Grid in 38.