On looking at the graph of a polynomial, we find that the graph cuts the x axis at certain points if it has real roots. We can use derivatives to find the roots of the equation formed from the polynomial.
In this method we first take a equation (f(x) = 0) whose root we have to find. Then we take a value x1. For that value of x we find the value (y1) of the corresponding expression. We find the derivative of the expression (f'(x)). Then we find the value of the slope (m1)at that point . We find the equation of a line
[(y − y1) = m (x − x1)].
Then we find the abscissa (x2) of intersection of the x-axis and the curve. We follow the same process again for the x2.
This method works very well for quadratic equation. It can work well for cubic equation and may work well for others. The problem which arises in this case is the value of x found. If the value of x found approaches the root then the method works well. There are many cases which arises when we find root by this method. The most favorable case is when we approach the root continuously i.e. every x with odd subscript moves in one direction only i.e. each successive odd term either becomes greater or either it becomes less of its previous. Each term with even subscript becomes greater if its corresponding even term is less and vice-verse. Then we will certainly approach the root.
Let us find the roots for a quadratic equation:
In this method we first take a equation (f(x) = 0) whose root we have to find. Then we take a value x1. For that value of x we find the value (y1) of the corresponding expression. We find the derivative of the expression (f'(x)). Then we find the value of the slope (m1)at that point . We find the equation of a line
[(y − y1) = m (x − x1)].
Then we find the abscissa (x2) of intersection of the x-axis and the curve. We follow the same process again for the x2.
This method works very well for quadratic equation. It can work well for cubic equation and may work well for others. The problem which arises in this case is the value of x found. If the value of x found approaches the root then the method works well. There are many cases which arises when we find root by this method. The most favorable case is when we approach the root continuously i.e. every x with odd subscript moves in one direction only i.e. each successive odd term either becomes greater or either it becomes less of its previous. Each term with even subscript becomes greater if its corresponding even term is less and vice-verse. Then we will certainly approach the root.
Let us find the roots for a quadratic equation:
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