First, we need to transform this into quadratic form by subtracting color(red)(25) from each side of the equation to keep the equation balanced while equating to 0:
9x^2 - color(red)(25) = 25 - color(red)(25)
9x^2 - 25 = 0
This is a special form of the quadratic equation which has the solution:
color(red)(a)x^2 - color(blue)(b) = (sqrt(color(red)(a))x + sqrt(color(blue)(b)))(sqrt(color(red)(a))x - sqrt(color(blue)(b))) = 0
Substituting from our quadratic gives:
color(red)(9)x^2 - color(blue)(25) = (sqrt(color(red)(9))x + sqrt(color(blue)(25)))(sqrt(color(red)(9))x - sqrt(color(blue)(25))) = 0
(3x + 5)(3x - 5) = 0
or, because the sqrt(9) = +-3
(-3x + 5)(-3x - 5) = 0
Now, we can solve each term for 0:
Solution 1)
3x + 5 = 0
3x + 5 - 5 = 0 - 5
3x + 0 = -5
3x = -5
(3x)/3 = -5/3
x = -5/3
Solution 2)
3x - 5 = 0
3x - 5 + 5 = 0 + 5
3x + 0 = 5
3x = 5
(3x)/3 = 5/3
x = 5/3
Solution 3)
-3x + 5 = 0
-3x + 5 - 5 = 0 - 5
-3x + 0 = -5
-3x = -5
(-3x)/(-3) = (-5)/-3
x = 5/3
Solution 4)
-3x - 5 = 0
-3x - 5 + 5 = 0 + 5
-3x + 0 = 5
-3x = 5
(-3x)/-3 = 5/-3
x = -5/3
x = 5/3 or x = -5/3
