soccer ball profit\n$y = -6x^{2}+100x - 180$\nsoccer balls go on sale for $7.50 each. the store also sells…

Answer:

Set the profit - equation for footballs equal to the target profit. Then solve the resulting quadratic equation (-4x^{2}+80x - 150=400) for (x). First, rewrite the equation in standard form (ax^{2}+bx + c = 0) as (-4x^{2}+80x-550 = 0) (by subtracting 400 from both sides). Divide through by - 2 to simplify: (2x^{2}-40x + 275=0). Then use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}), where (a = 2), (b=-40), and (c = 275). Calculate the discriminant (\Delta=b^{2}-4ac=(-40)^{2}-4\times2\times275=1600 - 2200=-600). Since the discriminant is negative, there are no real - valued solutions in this context, which means it may not be possible to meet the profit goal with the given profit model for footballs.

Explanation:

Step1: Set up the equation

Set (-4x^{2}+80x - 150 = 400) as the goal is to earn a profit of (y = 400).

Step2: Rewrite in standard form

(-4x^{2}+80x-150 - 400=0), so (-4x^{2}+80x - 550 = 0). Simplify to (2x^{2}-40x + 275 = 0).

Step3: Identify coefficients for quadratic formula

For (2x^{2}-40x + 275 = 0), (a = 2), (b=-40), (c = 275).

Step4: Calculate the discriminant

(\Delta=(-40)^{2}-4\times2\times275=1600 - 2200=-600).

Step5: Analyze the result

Since (\Delta<0), no real - valued solutions for (x).