3. [1/1 Points] DETAILS MY NOTES HARMATHAP12 2.3.003. If a company has total costs \( C(x)=45,000+55 x+0.1 x^{2} \) and total revenues given by \( R(x)=645 x-0.9 x^{2} \), find the break-even points. (Enter your answers as a comma-separated list.) \( x=\square \) D. [0/1 Points] DETAILS MY NOTES Solve. (Enter your answers as a comma-separated list.)

To find the break-even points, we need to set the total costs equal to the total revenues and solve for \( x \). Given: Total costs: \( C(x) = 45,000 + 55x + 0.1x^2 \) Total revenues: \( R(x) = 645x - 0.9x^2 \) Setting the total costs equal to the total revenues: \[ 45,000 + 55x + 0.1x^2 = 645x - 0.9x^2 \] Simplifying the equation: \[ 0.1x^2 + 0.9x^2 + 55x - 645x = 45,000 \] \[ 1x^2 - 590x + 45,000 = 0 \] Now, we can solve this quadratic equation to find the break-even points. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(1\times x^{2}-590x+45000=0\) - step1: Multiply: \(x^{2}-590x+45000=0\) - step2: Factor the expression: \(\left(x-500\right)\left(x-90\right)=0\) - step3: Separate into possible cases: \(\begin{align}&x-500=0\\&x-90=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=500\\&x=90\end{align}\) - step5: Rewrite: \(x_{1}=90,x_{2}=500\) The break-even points are \( x = 90 \) and \( x = 500 \).