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Transformations of Equations

If an equation is given, then it is possible for us to transform it into another such that the rots of the new equation have certain assigned relations with the roots of the given equation. The new equation is called transformed equation. The transformation of an equation sometimes facilitates in determining the roots of the given equation by solving the transformed equation, which is easier to solve. In this, section, we shall discuss the most important transformations. In the following article we shall assume that

To transform an equation into another whose roots are those of the proposed equation with contrary signs.

Let

Since

and

Also,

Therefore, equation of transformation is

Substituting this value of

Hence

Obviously, if

Then the required equation is

or,

_{1}**,**_{2}**, ….**_{n}are the roots of the given equation**ƒ(x) = 0 and**_{1}**,**_{2}**, …,**_{n}are the roots of the required equation.**Roots with signs changed**To transform an equation into another whose roots are those of the proposed equation with contrary signs.

Let

_{1}**,**_{2}**, ….**_{n}be the roots of the given equation and_{1}**,**_{2}**, …,**_{n}are the roots of the new equation**Ø(y) = 0**.Since

_{r}**=**_{r},and

**=**, satisfies**Ø(y) = 0**Also,

**x =**, satisfies**ƒ(x) = 0**Therefore, equation of transformation is

**y = -x**or**x = -y**Substituting this value of

**x**in**ƒ(x) = 0**, we get the transformed equation**ƒ(-y) = 0**.Hence

**Ø(y) = ƒ(-y)**.Obviously, if

**ƒ(x) = a**,_{0}x^{n}+ a_{1}x^{n-1}+ ….. + a_{n}= 0, a_{n}≠ 0Then the required equation is

**a**_{0}y^{n}– a_{1}y^{n-1}+ …. + (-1)^{n-1}a_{n-1}y + (-1)^{n}a_{n}= 0or,

**a**_{0}x^{n }– a_{1}x^{n-1}+ …. + (-1)^{n-1}a_{n-1}x + (-1)^{n}a_{n}= 0.**Working rule:**The transformed equation is obtained from the given equation by changing the signs of every alternate coefficient beginning with the second after making the given equation complete in case it is not complete.**Services: -**Transformations of Equations Homework | Transformations of Equations Homework Help | Transformations of Equations Homework Help Services | Live Transformations of Equations Homework Help | Transformations of Equations Homework Tutors | Online Transformations of Equations Homework Help | Transformations of Equations Tutors | Online Transformations of Equations Tutors | Transformations of Equations Homework Services | Transformations of EquationsSubmit Your Query ???

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