Solve the equation (x/5 + 1) times (x/2) = 2488.

Step-by-step solution

1. 1

   Multiply out the product. Distributing the $\frac{x}{2}$ over the bracket,

   $\left(\frac{x}{5}+1\right)\frac{x}{2}=\frac{x^2}{10}+\frac{x}{2}$

   so the equation is quadratic, not linear — there will be two roots.

2. 2

   Clear the denominators. The least common denominator of $10$ and $2$ is $10$, so multiply everything by $10$:

   $x^2+5x=24880$

3. 3

   Put it in standard form.

   $x^2+5x-24880=0$

4. 4

   Compute the discriminant and check whether the root simplifies.

   $\Delta=5^2-4(1)(-24880)=25+99520=99545$

   Factoring, $99545=5\cdot 43\cdot 463$ — three distinct primes, no square factor — so $\sqrt{99545}$ cannot be simplified. Numerically $\sqrt{99545}=315.50753$.

5. 5

   Apply the quadratic formula.

   $x=\frac{-5\pm\sqrt{99545}}{2}\quad\Rightarrow\quad x_1=155.2538,\quad x_2=-160.2538$

   Rounding to $155.20$ is not accurate enough: substituting it gives $\left(\frac{155.20}{5}+1\right)\frac{155.20}{2}=2486.30$, missing the target by about $1.7$.

6. 6

   Verify with Vieta and by substitution. The roots sum to $-5$ and multiply to $-24880$, matching $-b/a$ and $c/a$ ✓. And directly, $\left(\frac{155.2538}{5}+1\right)\frac{155.2538}{2}=2488.00$ ✓. If the context requires a positive quantity, take $x\approx 155.25$.