Solution of Equation in One Variable

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.

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.

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.

Digital STUDY Center

Digital Study Center offers an effective and amazing learning platform for keen learn students in the world. We identify the needs and demands of the keen learn students which is why we stand out unique in the crowd.

Post A Comment:


Dear readers,
Your feedback is usually appreciated. We'll reply to your queries among 24hrs. Before writing your comments, please read the the subsequent directions attentively:

1. Please comments in English. We accept only English comments.

2. Please don't Spam. All spammed comments will be deleted as before long as pobile, after review.

3. Please don't Add Links with your comments as they won't be published.

4. If We can be of assistance, please don't hesitate to contact us.