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Home » Math Homework Help » Algebra Homework Help » Transformations of Equations
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 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) = a0xn + a1xn-1 + ….. + an = 0, an ≠ 0,

Then the required equation is

a0yn – a1yn-1 + …. + (-1)n-1 an-1 y + (-1)n an = 0

or, a0xn – a1xn-1 + …. + (-1)n-1 an-1 x + (-1)n an = 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.

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