Solve for x: x(-5x + 250) = 2625.

Step-by-step solution

1. 1

   Recognise the shape. A product like $x(-5x+250)$ is the classic revenue or area model: $x$ units at a price that falls as $x$ rises. Setting it equal to $2625$ asks which two quantities produce that target — expect two answers, one on each side of the maximum.

2. 2

   Expand and move everything to one side.

   $-5x^2+250x=2625\qquad\Longrightarrow\qquad -5x^2+250x-2625=0$

3. 3

   Divide by -5 to make the leading coefficient 1. This is the step that turns ugly numbers into factorable ones. Dividing an equation (not an inequality) by a negative is harmless:

   $x^2-50x+525=0$

   Note $250/(-5)=-50$ and $-2625/(-5)=+525$ — both signs flip.

4. 4

   Factor the monic quadratic. Look for two numbers multiplying to $+525$ and summing to $-50$; both must be negative. Since $525=3\cdot5^2\cdot7$, the candidate pairs are $(1,525),(3,175),(5,105),(7,75),(15,35),(21,25)$, and $15+35=50$:

   $x^2-50x+525=(x-15)(x-35)$

5. 5

   Apply the zero-product property.

   $x-15=0\Rightarrow x=15,\qquad x-35=0\Rightarrow x=35$

6. 6

   Verify both in the original equation. At $x=15$: $15(-75+250)=15\times175=2625$ ✓. At $x=35$: $35(-175+250)=35\times75=2625$ ✓. As a structural check, the roots sum to $50=-b/a$ and multiply to $525=c/a$ ✓, and their midpoint $25$ is where the parabola peaks at $25\times125=3125$ — comfortably above $2625$, as it must be for two solutions to exist.