Instruction
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.