Solve 3600x + 2430000 = x2.

Step-by-step solution

1. 1

   Rearrange into standard form. Move everything to the side where $x^2$ is positive:

   $x^2-3600x-2430000=0$

   so $a=1$, $b=-3600$, $c=-2430000$. Because $c<0$ the two roots will have opposite signs.

2. 2

   Compute the discriminant.

   $\Delta=(-3600)^2-4(1)(-2430000)=12960000+9720000=22680000$

3. 3

   Simplify the square root instead of decimalising it. Strip out perfect squares:

   $22680000=324\cdot 70000=324\cdot 10000\cdot 7$

   so

   $\sqrt{22680000}=18\cdot 100\cdot\sqrt7=1800\sqrt7$

   (Confirm: $(1800)^2\cdot 7=3240000\cdot 7=22680000$ ✓.) Keeping the surd is what makes the final answer exact.

4. 4

   Apply the quadratic formula.

   $x=\frac{3600\pm 1800\sqrt7}{2}=1800\pm 900\sqrt7$

   Both terms are halved — a very common slip is to halve only the $3600$.

5. 5

   Evaluate numerically. With $\sqrt7=2.6457513$, $900\sqrt7=2381.1762$, so

   $x_1=4181.1762,\qquad x_2=-581.1762$

   Opposite signs, as step 1 predicted.

6. 6

   Check with Vieta. The roots must sum to $-b/a=3600$ and multiply to $c/a=-2430000$. Indeed $(1800+900\sqrt7)+(1800-900\sqrt7)=3600$ ✓, and

   $(1800)^2-(900\sqrt7)^2=3240000-5670000=-2430000$ ✓