The height of a triangle is 25 inches greater than its base. The area of the triangle is 625 square inches. The height of the triangle is inches and the base of the triangle is inches.

base --- x

height --- x+25

(1/2)(x)(x+25) = 625
x^2 + 25x = 1250
x^2 + 25x - 1250 = 0
(x-25)(x+50) = 0
x = 25 or an inadmissible negative

To find the height and base of the triangle, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Given that the area is 625 square inches, we can substitute this value into the formula:

625 = (1/2) * base * height

Now, let's use the information that the height is 25 inches greater than the base. We can express this as an equation:

height = base + 25

Substituting this expression for the height into the area formula, we get:

625 = (1/2) * base * (base + 25)

Next, let's solve this equation for the base. Multiply everything by 2 to get rid of the fraction:

1250 = base * (base + 25)

Simplifying, we have:

1250 = base^2 + 25base

Let's rearrange this equation to form a quadratic equation:

base^2 + 25base - 1250 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. I will use the quadratic formula to find the values of the base.

The quadratic formula is given as:

base = (-b ± √(b^2 - 4ac)) / (2a)

For our quadratic equation, the coefficients are:
a = 1
b = 25
c = -1250

Substituting these values into the quadratic formula, we get:

base = (-25 ± √(25^2 - 4 * 1 * -1250)) / (2 * 1)

Simplifying further, we have:

base = (-25 ± √(625 + 5000)) / 2

base = (-25 ± √5625) / 2

base = (-25 ± 75) / 2

From this equation, we get two possible values for the base:

base = (-25 + 75) / 2 or base = (-25 - 75) / 2

Simplifying, we have:

base = 50 / 2 or base = -100 / 2

base = 25 or base = -50

Since the base of a triangle cannot be negative, we can discard the solution base = -50. Therefore, the only valid solution is:

base = 25 inches

Now, let's use this value to find the height. We know that the height is 25 inches greater than the base:

height = base + 25

Substituting the value for the base, we get:

height = 25 + 25

height = 50 inches

Therefore, the height of the triangle is 50 inches and the base is 25 inches.