Find the root of the equation x^2 - 15 = (x + 15)^2.

Step-by-step solution

1. 1

   Expand the right-hand side.

   $(x+15)^2 = x^2 + 30x + 225$

   so the equation reads $x^2 - 15 = x^2 + 30x + 225$.

2. 2

   Cancel the quadratic terms. Both sides carry exactly one $x^2$, so subtracting $x^2$ removes it entirely:

   $-15 = 30x + 225$

   The equation is therefore linear and has a single root — despite the squares on both sides suggesting two.

3. 3

   Solve for x.

   $30x = -15 - 225 = -240 \quad\Longrightarrow\quad x = -8$

4. 4

   Check by substitution. Left: $(-8)^2 - 15 = 64 - 15 = 49$. Right: $(-8 + 15)^2 = 7^2 = 49$. Both give $49$.

5. 5

   Interpret geometrically. The equation asks where the parabola $y = x^2$ shifted down $15$ meets the same parabola shifted left $15$. Two congruent parabolas differing only by a translation cross in exactly one point — here $(-8, 49)$ — which is the structural reason only one root exists.