# When all the terms of a quadratic equation are increased by a particular number do the roots remain the same?

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Let’s take a quadratic equation ax^2 + bx + c = 0

The roots of the equation are given by x1 = [-b + sqrt (b^2 + 4ac)]/ 2a and x2 = [-b - sqrt (b^2 + 4ac)]/ 2a

Now if all the terms of the equation are increased by a particular number D, the equation becomes ax^2 + D + bx + D + c + D =0

=> ax^2 + bx + c + 3D =0

=> ax^2 + bx + c + 3D

Now if we find the roots of this equation we see that the coefficient c has changed to c + 3D. This cannot give the same roots as ax^2 + bx + c = 0

The roots remain the same only if each of the terms is multiplied or divided by a common term as then the common term can be taken as a common factor and eliminated.

Let the quadratic equation be ax^2+bx+c =0.

Then the roots are given by:

x1 = {-b+sqrt(b^2-4ac)}/2a......(1)

x2 = {-b-sqrsqrt(b^2-4ac)}/2a.......(2)

There are 3 terms in the quadratic equation ax^2+bx+c = 0.

now we increase each term by a particular value k. Then the quadratic equation becomes:

(ax^2+k)+(bx+k)+(c+k) = 0.

ax^2+bx +(c+3k) = whose roots are given by:

x1 = {-b+sqrt(b^2-4a(c+3k))}/2a......(3).

x2 = {-b-sqrt(b^2-4a(c+3k))}/2a.......(4).

Clearly, the roots at (3) and (4)after increasing each term of the quadratic equation by a particular value k is different from the roots at (2) and(3).