To solve the equation
[tex]\[
\frac{1500}{x} - \frac{1}{2} = \frac{1500}{x + 250},
\][/tex]
we will go through a step-by-step process.
1. Eliminate the fractions by finding a common denominator:
Let's rewrite the equation to have a common denominator:
[tex]\[
\frac{1500}{x} - \frac{1500}{x+250} = \frac{1}{2}.
\][/tex]
2. Clear the fractions by multiplying through by the common denominator:
The common denominator here is [tex]\(2x(x+250)\)[/tex]:
[tex]\[
2(x+250) \cdot 1500 - 2x \cdot 1500 = x(x+250).
\][/tex]
Simplify each term:
[tex]\[
3000(x + 250) - 3000x = x(x + 250).
\][/tex]
3. Distribute and combine like terms:
[tex]\[
3000x + 750000 - 3000x = x^2 + 250x.
\][/tex]
Notice the [tex]\(3000x\)[/tex] terms cancel out:
[tex]\[
750000 = x^2 + 250x.
\][/tex]
4. Rearrange the equation into standard quadratic form:
[tex]\[
x^2 + 250x - 750000 = 0.
\][/tex]
5. Solve the quadratic equation:
The quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] can be solved using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 250\)[/tex], and [tex]\(c = -750000\)[/tex]:
[tex]\[
x = \frac{-250 \pm \sqrt{250^2 - 4 \cdot 1 \cdot (-750000)}}{2 \cdot 1}.
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-250 \pm \sqrt{62500 + 3000000}}{2}.
\][/tex]
[tex]\[
x = \frac{-250 \pm \sqrt{3062500}}{2}.
\][/tex]
Simplify the square root:
[tex]\[
x = \frac{-250 \pm 1750}{2}.
\][/tex]
6. Find the potential solutions:
[tex]\[
x = \frac{-250 + 1750}{2} \quad \text{and} \quad x = \frac{-250 - 1750}{2}.
\][/tex]
These give us:
[tex]\[
x = \frac{1500}{2} = 750 \quad \text{and} \quad x = \frac{-2000}{2} = -1000.
\][/tex]
So, the solutions to the equation [tex]\(\frac{1500}{x} - \frac{1}{2} = \frac{1500}{x + 250}\)[/tex] are
[tex]\[
x = 750 \quad \text{and} \quad x = -1000.
\][/tex]
