In a math class, the quadratic x^2+10x+20 is written on the board. Each student goes to the board and increases or decreases either the linear or constant coefficient by 1. After some time, x^2+20x+10 is written on the board. Did a quadratic with integer roots necessarily appear on the board at some time during this process? Why or why not? Pleas explain in detail!!!
Not sure EXACTLY what your parmeters are in your question, but look at this scenario:
x^2+10x+ 20 first student increases the x coefficient by ONE
x^2 + 11x + 20 the next 10 students DECREASE the constant by ONE:
x^2 + 11x + 10 the next 9 students INCREASE the x coefficient by ONE
x^2 + 20x + 10 The end
the RED intermediate quadratic is factorable to (x+1)(x+10) two integer roots of -1 and -10
Is THAT what you meant???
Updated perspective: If we can keep the disciminant (b^2-4ac) from being ZERO or a perfect square the quadratic will not have integer roots...... I tried to 'brute force' my way through the combinatins, but it was tedious (kept running in to integer roots)....there must be another way...an Excel spreadsheet would be nice ! LOL
I worked on this some and I think it is impossible to go from the beginning equation to the final equation without writing down an equation with integer roots at least once. At some point the constant, "c", will be exactly one greater than the linear coefficient, "b". When c is one greater than b, there will be integer roots.