Solve the equation by factoring, show all work.

25x^2-30x+9=0

we know 5*3 is 15

and two 15s is 30
so

(5x-3)(5x-3) = 0
so
5x-3 = 0
x = 3/5 twice because both solutions are the same. That means the vertex of the parabola is on the x axis at x = 5

Use AC method of factoring.

25x^2-30x+9 = 0.
A*C = 25*9 = 225 = 3*75 = (-15)*(-15).
Select the set of factors whose sum = -30.

25x^2 + (-15x+(-15x)) + 9 = 0.
(25x^2-15x) + (-15x+9) = 0,
5x(5x-3) + -3(5x-3) = 0,
(5x-3)(5x-3) = 0,
5x-3 = 0, X = 3/5.

To solve the given equation by factoring, we need to find two binomials that multiply together to give us the quadratic expression in the equation. Let's begin:

1. Start by inspecting the constant term, which is 9 in this case. We want to find two numbers that multiply together to give 9.
The possibilities are:
1 * 9 = 9
-1 * -9 = 9
3 * 3 = 9
-3 * -3 = 9

2. Now we look at the coefficient of the middle term, which is -30. We want to find two numbers that add up to give -30.
The possibilities are:
(1 + 9) = 10
(-1 + -9) = -10
(3 + 3) = 6
(-3 + -3) = -6

3. Since our desired sum is -30, we can see that -3 and -3 are the values we need. Therefore, we can rewrite the middle term (30x) as -3x - 27x. Now our equation looks like this:
25x^2 - 3x - 27x + 9 = 0

4. Group the terms and factor by grouping:
(25x^2 - 3x) + (-27x + 9) = 0
x(25x - 3) - 9(25x - 3) = 0

5. Notice that we have a common binomial factor, 25x - 3. We can factor it out:
(x - 9)(25x - 3) = 0

6. Now, we need to set each factor equal to zero and solve for x:
x - 9 = 0 or 25x - 3 = 0

For the first factor:
x = 9

For the second factor:
25x = 3
x = 3/25

7. Therefore, the solutions to the equation 25x^2 - 30x + 9 = 0 are x = 9 and x = 3/25.