Im having trouble factoring 25x^2-10x+1 using the ac method. I end up with (5x-5)(5x-5)

Question

Im having trouble factoring 25x^2-10x+1 using the ac method. I end up with (5x-5)(5x-5)

Answer 1

 Have to choose 2 numbers whose some equals -10, and product 25

 

   How about -5, -5    a+b = -10

                                a.b = 25     So the factors are 

 

 

      -10X = -5X - 5X

 

    25 X² - 5X -5X +1 =

   5X ( 5X-1 )  - ( 5X -1) =

    (5X -1 ) ( 5X - 1) = 25X² - 10X +1

 

    You should be good in the quadratic expression: 

   

     From ( a ±b )² = a²±2ab + b²    and  recognize this immediately.

 

 

 

 

 

 

 

Answer 2

(a ± b) = a² ± 2ab + b²
~~~~~~~~~~~~
25x² - 10x + 1 = (5x)² - 2· 5x · 1 + 1² = (5x - 1)² 

In general:
ax² + bx + c = a(x - x₁)(x - x₂) , x₁ and x₂ are roots of equation ax² + bx + c = 0 , and
x₁₂ = (-b ± √D)/2a , D = b² - 4ac .
~~~~~~~~~~~~
25x² - 10x + 1 = 0
D = (-10)² - 4 · 25 · 1 = 100 - 100 = 0
the equation has only one root (actually 2 roots with the same value) 
x₁₂ = -(-10)/(2 · 25)
x₁₂ = 1/5
25(x - 1/5)(x - 1/5) = (5x - 1)(5x - 1) = (5x - 1)²