aleks answers - 1. Finding the roots of a quadratic...

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1. Finding the roots of a quadratic equation with leading coefficient greater than 1 Solve: .
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We first rewrite as . Factoring the left-hand side of this equation, we obtain . Note that in this last equation we have the product of two expressions, and , equal to . This can be true only if at least one of the expressions equals . So we must have or . Solving these two equations for yields or . The answer is , . . Augmented matrix and solution set of a system of linear equations
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Suppose that the augmented matrix of a system of linear equations for unknowns , , and is . Solve the system and provide the information requested. Background: As in the case of systems of two equations in two unknowns , any system of linear equations is one of the following three types: 1. The system has a unique solution. In this case we say that the system is consistent independent . 2. The system has infinitely many solutions. Such a system is called consistent dependent . 3. The system has no solution. In this case we call the system inconsistent . Here we will consider the case of systems of three equations in three unknowns. More The type of a system and all its solutions (if they exist) can be determined by transforming the augmented matrix of the system into reduced row-echelon form . Denoting the augmented matrix in the reduced row-echelon form by , we have the following: If has a row consisting of a nonzero entry in the column of constant terms but zeros otherwise, then the system is inconsistent. For example, if the reduced row-echelon form of the augmented matrix of a system is ,
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then the system is inconsistent. Indeed, the equation corresponding to the last row is , which cannot be satisfied for any values of , , and . If no such row in exists, then the system is consistent. To determine whether the system is consistent dependent or consistent independent, we examine whether there are rows of with all entries equal to . 1. If there are no rows with all entries equal to , the system is consistent independent. For example, if the reduced row-echelon form of the augmented matrix is , then the system is consistent independent. In particular, it has the unique solution . 2. If there are rows with all entries equal to , then the system is consistent dependent. In this case unknowns can be arbitrary real numbers, called parameters . The remaining unknowns can then be expressed in terms of these parameters. For example, if the reduced row-echelon form of the augmented matrix is , then , and the system is consistent dependent. Because , one unknown can be an arbitrary number while the other two unknowns can be found from the system. Namely, the system of equations in this case is .
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Letting
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aleks answers - 1. Finding the roots of a quadratic...

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