GMAT Practice Question: The set of solutions for the equation (x^{2}-25)^{2}=x^{2}-10 x+25 contains how many real numbers?
Question
The set of solutions for the equation (x^{2}-25)^{2}=x^{2}-10 x+25 contains how many real numbers?
- 0
- 1
- 2
- 3
- 4
Topics: quadratic equations, factoring, development formulas
Solution
Step 1
We notice that the right-hand side is the square of x minus 5.
(x^2 - 25)^2 = (x - 5)^2
Step 2
We move all terms to one side to create a difference of squares.
(x^2 - 25)^2 - (x - 5)^2 = 0 ((x^2 - 25) - (x - 5))\times((x^2 - 25) + (x - 5)) = 0
Step 3
We set the first factor equal to zero and solve for x.
((x^2 - 25) - (x - 5)) = 0 x^2 - x - 20 = 0 (x-5)(x+4) = 0 x = 5, -4
Step 4
We set the second factor equal to zero and solve for x.
((x^2 - 25) + (x - 5)) = 0 x^2 + x - 30 = 0 (x+6)(x-5) = 0 x = -6, 5
Step 5
We collect the distinct real solutions.
x = -6, -4, 5
Answer
3
