To solve the quadratic equation 25x² = 20x - 4 by factoring, follow these steps:
Start by bringing all terms to one side of the equation to set it equal to zero:
25x² - 20x + 4 = 0
Now, attempt to factor the quadratic expression. Look for two numbers that multiply to the coefficient of the leading term (25) times the constant term (4), which is 100. These numbers should also add up to the coefficient of the middle term (-20).
The two numbers that fit these criteria are -10 and -10, because (-10) * (-10) = 100 and (-10) + (-10) = -20.
Use these numbers to factor the quadratic:
25x² - 20x + 4 = 25x² - 10x - 10x + 4
Factor by grouping:
(25x² - 10x) - (10x - 4)
Factor out the greatest common factor from each group:
5x(5x - 2) - 2(5x - 2)
Notice that both terms now have a common factor of (5x - 2):
(5x - 2)(5x - 2)
Rewrite it as a binomial square:
(5x - 2)² = 0
Now, set each factor equal to zero and solve for x:
5x - 2 = 0
5x = 2
x = 2/5
So, the solutions to the equation are x = 2/5, and it's a double root, which means there's only one unique solution: x = 2/5.