--> F(x) = xsud21 wrote:F(x)=25X^3-x, if F(x)=x, what is sum of all solutions of x?
--> 25x³ - x = x
--> 25x³ - 2x = 0
Now this is a cubic equation which has three roots.
Let's say three roots of the above equation are a, b, and c.
Hence, the equation can be written as (x - a)(x - b)(x - c) = 0
Multiplying the terms, (x³ - (a + b + c)x² + (ab + bc + ca)x - abc) = 0
Now comparing with the original equation we have, we can see that the co-efficient of x² is zero. Hence, the sum of the roots, i.e. (a + b + c) = 0
Another method to solve this problem
--> 25x³ - 2x = 0
--> x(25x² - 2) = 0
--> x[(5x)² - (√2)²] = 0
--> x(5x - √2)(5x + √2) = 0
Hence, the roots of the equation are 0, -√2/5, and √2/5.
Hence, sum of the roots = (0 + √2/5 - √2/5) = 0
