Question

let x be the number of $1 decreases let r be revenue r = (500 + 20x)(75 - x) 22500 - 500x + 1500x - 20x²

Explanation:

The handwriting in the image is a bit unclear, but let's assume we are trying to solve an equation related to revenue or profit (common in Business - Finance or Mathematics - Algebra). Let's try to parse the left - most expression: \(22500 - 500x+400x - 20x^{2}\). First, combine like terms. The terms with \(x\) are \(- 500x\) and \(+400x\).

Step 1: Combine like terms for \(x\)

\(-500x + 400x=-100x\). So the expression becomes \(22500-100x - 20x^{2}\). We can also factor out a \(- 20\) to simplify further.

Step 2: Factor out - 20

\(22500-100x - 20x^{2}=-20x^{2}-100x + 22500=-20(x^{2}+5x - 1125)\)
If we assume this is a quadratic equation set to zero (for example, if we are finding the break - even point in a business context or solving for \(x\) in a mathematical equation), we can use the quadratic formula. For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions are \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In the equation \(x^{2}+5x - 1125 = 0\), \(a = 1\), \(b = 5\), and \(c=-1125\).

Step 3: Calculate the discriminant

The discriminant \(D=b^{2}-4ac=(5)^{2}-4\times1\times(-1125)=25 + 4500 = 4525\)

Step 4: Find the square root of the discriminant

\(\sqrt{4525}\approx67.27\)

Step 5: Solve for \(x\)

\(x=\frac{-5\pm67.27}{2}\)
For the plus sign: \(x=\frac{-5 + 67.27}{2}=\frac{62.27}{2}=31.135\)
For the minus sign: \(x=\frac{-5-67.27}{2}=\frac{-72.27}{2}=-36.135\)

Since in most business or real - world contexts (like the number of items sold), \(x\) should be non - negative, we take \(x\approx31.14\) (if we are solving for a variable related to quantity).

Answer:

If we are solving \(22500 - 500x + 400x-20x^{2}=0\) (assuming it's set to zero for a break - even or solution), the positive solution for \(x\) is approximately \(31.14\) (the exact form using the quadratic formula is \(x=\frac{-5\pm\sqrt{4525}}{2}\))