Question

x⁴-5x³+8x²-5x+1=0 find the roots​

Answer 1

Step-by-step explanation:

Solve for x:

x^4 - 5 x^3 + 8 x^2 - 5 x + 1 = 0

The left hand side factors into a product with two terms:

(x - 1)^2 (x^2 - 3 x + 1) = 0

Split into two equations:

(x - 1)^2 = 0 or x^2 - 3 x + 1 = 0

Take the square root of both sides:

x - 1 = 0 or x^2 - 3 x + 1 = 0

Add 1 to both sides:

x = 1 or x^2 - 3 x + 1 = 0

Subtract 1 from both sides:

x = 1 or x^2 - 3 x = -1

Add 9/4 to both sides:

x = 1 or x^2 - 3 x + 9/4 = 5/4

Write the left hand side as a square:

x = 1 or (x - 3/2)^2 = 5/4

Take the square root of both sides:

x = 1 or x - 3/2 = sqrt(5)/2 or x - 3/2 = -sqrt(5)/2

Add 3/2 to both sides:

x = 1 or x = 3/2 + sqrt(5)/2 or x - 3/2 = -sqrt(5)/2

Add 3/2 to both sides:

Answer: |

| x = 1 or x = 3/2 + sqrt(5)/2 or x = 3/2 - sqrt(5)/2