Solution of Equation in One Variable

Introduction:
The equation of the form f(x)=0 is called an equation in one variable. It may be algebraic or transcendental or a combination of both. When f(x) is a polynomial in x, that is f(x)=a0xn+a1xn-1++a2xn-2+ ..... +an then f(x)=0 is called algebraic equation, Again if f(x) contains some other functions such as logarithmic, exponential, trigonometric etc, then f(x)=0 is called transcendental equation.

Example:
A few one variable equation:
(i) x3-2x-5=0
(ii) x3+x2-1=0
(iii) x2-3=0 etc.

A few one variable equation:
(i) e-x=sinx
(ii) etanx=1
(iii) 2x=log10x+7 etc.
If  f(x)=0 is an equation and f(a)=0 then x=a is called a root of the equation. Again if the graph of y=f(x) crosses the x-axis at x=a then x=a is a root of the equation f(x)=0. Sometimes root of equation is called zero or solution of the equation.

Mathematical Example:
f(x)=x3-6x2+11x-6=0 is an equation and f(1)=1-6+11-6=0, So x=1 is a root of the equation. Similarly x=2 and x=3 are roots of the equation.

Graph: f(x)=x3-6x2+11x-6=0

The graph of f(x)=x3-6x2+11x-6 crosses the x-axis at (-2, 0) and (3,0). So x=-2 and x=3 are the roots of the equation f(x)=0.

Graphical Method

Discuss the Graphical method to find a real root of an equation f(x)=0 

Solution: If we take a set of rectangular coordinate axes and plot the graph of y=f(x), then the abscissas of the point where the graph crosses the x-axis are the real roots of the equations f(x)=0.

In most cases the approximate values of the real roots of f(x)=0 are most easily found by writing the equation in the form f1(x)=f2(x) and then plotting the two equations y=f1(x) and y=f2(x) on the same axes. The abscissas of the point of intersection of these two functions y=f1(x) and y=f2(x) are the real roots of the equation f(x)=0.

Problem: Find the approximate value of a root of the equation cosx=3x-1 usnig graphical method.

Solution:
Given equation, cosx=3x-1
Let, y=cosx ............................ (i)
and y=3x-1 ............................ (ii)

Now we plot (i) and (ii) separately on the same set of axes as shown on the following diagram.

Graph: cosx=3x-1

The abscissa of the point of intersection not the graphs of theses equations is seen to be about 0.6. hence the approximate value of the root is 0.6.
Axact

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