In business, the break-even point is the point (x, y) at which the graphs of the revenue and cost functions intersect. This is the point where the revenue and the expenses are equal. The business is not losing money, but it is not making money either. For one manufacturing company, the revenue from producing x items is given by the function y = 2x + 12 and the cost of producing x items is given by y = −x2 + 10x + 5. The numbers represent thousands. Find all break-even points. Interpret the results.

The break-even points are (1 , 14) and (7 , 26), the numbers in thousandsWhen the company produces 1 thousand items the revenue and expenses are equal 14 thousands When the company produces 7 thousands items the revenue and the expenses are equal 26 thousandsStep-by-step explanation:The break-even point is the point (x, y) at which the graphs of therevenue and cost functions intersectFor one manufacturing company, the revenue from producing x items is given by the function y = 2x + 12, and the cost ofproducing x items is given by y = -x² + 10x + 5The numbers represent thousandsWe need to find all break-even pointsTo solve the problem do that:1. equate the two equations2. Collect all terms in one side and put the other side 03. Factorize the quadratic to find the values of x4. substitute the value of x in the linear equation to find the values of y∵ y = 2x + 12∵ y = -x² + 10x + 5∴ 2x + 12 = -x² + 10x + 5- Add x² to both sides∴ x² + 2x + 12 = 10x + 5- Subtract 10x from both sides∴ x² - 8x + 12 = 5- Subtract 5 from both sides∴ x² - 8x + 7 = 0- Factorize the left hand side∴ (x - 1)(x - 7) = 0- Equate each bracket by 0∵ x - 1 = 0- Add 1 to both sides∴ x = 1∵ x - 7 = 0- Add 7 to both sides∴ x = 7∴ x = 1 and x = 7Substitute the values of x in the equation y = 2x + 12 to find the values of y∵ y = 2x + 12∵ x = 1∴ y = 2(1) + 12∴ y = 14∵ x = 7∴ y = 2(7) + 12∴ y = 14 + 12∴ y = 26∴ y = 14 and y = 26∴ The break-even points are (1 , 14) and (7 , 26), the numbers in    thousandsThe break-even points are (1 , 14) and (7 , 26), the numbers in thousandsWhen the company produces 1 thousand items the revenue and expenses are equal 14 thousands When the company produces 7 thousands items the revenue and the expenses are equal 26 thousandsLearn more;You can learn more about quadratic equations in brainly.com/question/1357167#LearnwithBrainly