The simplest type of tasks with parameters – problem on square trinomial A·x2+B·x+C Parametric value can be any of the coefficients of the equation: A, B or C. to Find the roots of trinomial square for every parameter value – so to solve the quadratic equation A·x2+B·x+C=0, listing each of the possible values of the unfixed variables.
In principle, if the equation A·x2+B·x+C=0 is a parameter of the senior coefficient A, it will be a square only when A≠0. When A=0 it degenerates into the linear equation B·x+C=0 has one root: x=-C/B. Therefore, test condition A≠0, A=0 must be the first line item.
The quadratic equation has real roots with non-negative discriminant D=B2-4·A·C If D>0 it has two distinct roots, if D=0 only one. Finally, if D
Often for solving problems with the settings, use the theorem of vieta. If the quadratic equation A·x2+B·x+C=0 has roots x1 and x2, the true system: x1+x2=-B/A x1·x2=C/A. the quadratic equation leading coefficients equal to one, we see that: x2+M·x+N=0. For him, the vieta theorem has a simplified form: x1+x2=-M, x1·x2=N. it Should be noted that the vieta theorem is true in the presence of one or two roots.
The same roots are found by using the vieta theorem, we can substitute back into the recording equation: x2-(x1+x2)·x+x1·x2=0. Do not confuse: here x is a variable, x1, and x2 - specific number.
Often helps in deciding the method of factorization. Let the equation A·x2+B·x+C=0 has roots x1 and x2. When the true identity of A·x2+B·x+C=A·(x-x1)·(x-x2). If the root only, then you can just say that x1=x2, then A·x2+B·x+C=A·(x-x1)2.
Example. Find all integers p and q, in which the roots of the equation x2+p·+q=0 is equal to p and q.Solution. Let p and q satisfy the conditions of the problem, that is, they are roots. Then by vieta theorem:p+q=-p,pq=q.
The system is equivalent to the conjunction p=0, q=0 or p=1, q=-2. It now remains to check is to make sure that the resulting numbers do satisfy the problem statement. For this you just need to substitute the numbers into the original equation.Answer: p=0, q=0 or p=1, q=-2.