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Transformation in General

Let

form an equation whose roots are

Now, , , are the roots of the equation

∴ = r

Since

Or, ry

Or, rx

is the required equation.

**ƒ(x) = 0**be a given equation whose roots are connected by the given relation**F(x, y) = 0**, known as the equation of transformation. Then the transformed equation**Ø(y) = 0**is obtained by eliminating**x**from**ƒ(x) = 0**and**F(x, y) = 0**.**Example:**if , , are the roots of the equation**x**,^{3}– px^{2}+ qx = 0form an equation whose roots are

Now, , , are the roots of the equation

**x**^{3}– px^{2}+ qx – r = 0 (1)∴ = r

Since

**x =**is the root of the equation**(1)**, hence the relation of transformation isOr, ry

^{3}– q(1 + r)y^{2}+ p(1 + r)^{2}y – (1 + r)^{3}= 0Or, rx

^{3}– q(1 + r)x^{2}+ p(1 + r)x = (1 + r)^{3}= 0is the required equation.

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