(80-x)(500+20x)=55000

(80-x)(500+20x)=55000

The correct answer and explanation is:

Let’s solve the equation (80−x)(500+20x)=55000(80 – x)(500 + 20x) = 55000.

Step 1: Expand the equation

First, apply the distributive property to expand the left-hand side: (80−x)(500+20x)=80⋅500+80⋅20x−x⋅500−x⋅20x(80 – x)(500 + 20x) = 80 \cdot 500 + 80 \cdot 20x – x \cdot 500 – x \cdot 20x

Now simplify each term: =40000+1600x−500x−20×2= 40000 + 1600x – 500x – 20x^2

Combine like terms: =40000+1100x−20×2= 40000 + 1100x – 20x^2

So the equation becomes: 40000+1100x−20×2=5500040000 + 1100x – 20x^2 = 55000

Step 2: Rearrange the equation

To solve for xx, move all terms to one side of the equation: 40000+1100x−20×2−55000=040000 + 1100x – 20x^2 – 55000 = 0

Simplify: −15000+1100x−20×2=0-15000 + 1100x – 20x^2 = 0

Multiply through by -1 to make the leading coefficient positive: 15000−1100x+20×2=015000 – 1100x + 20x^2 = 0

Step 3: Solve the quadratic equation

Now, we have a quadratic equation: 20×2−1100x+15000=020x^2 – 1100x + 15000 = 0

To simplify further, divide the entire equation by 20: x2−55x+750=0x^2 – 55x + 750 = 0

Use the quadratic formula to solve for xx: x=−(−55)±(−55)2−4(1)(750)2(1)x = \frac{-(-55) \pm \sqrt{(-55)^2 – 4(1)(750)}}{2(1)} x=55±3025−30002x = \frac{55 \pm \sqrt{3025 – 3000}}{2} x=55±252x = \frac{55 \pm \sqrt{25}}{2} x=55±52x = \frac{55 \pm 5}{2}

So, there are two possible solutions: x=55+52=30orx=55−52=25x = \frac{55 + 5}{2} = 30 \quad \text{or} \quad x = \frac{55 – 5}{2} = 25

Step 4: Conclusion

The two possible values for xx are 3030 and 2525.