The height of a triangle is 5 units more than its base. The area of the triangle is 18 units less than the square of the base. Find the area of the triangle

To solve this problem, we need to use the formula for the area of a triangle, which is 1/2 * base * height. We also know that the height is 5 units more than the base, and the area is 18 units less than the square of the base.

Let's denote the base as x. Then the height is x + 5.

According to the problem, the area of the triangle is x^2 - 18.

We can set up the equation as follows:

1/2 * x * (x + 5) = x^2 - 18

Solving this equation will give us the value of x, which we can then use to find the area of the triangle.

First, let's simplify the equation:

1/2 * x^2 + 5/2 * x = x^2 - 18

Subtract 1/2 * x^2 from both sides:

5/2 * x = 1/2 * x^2 - 18

Multiply both sides by 2 to get rid of the fraction:

5x = x^2 - 36

Rearrange the equation to set it equal to zero:

x^2 - 5x - 36 = 0

This is a quadratic equation, which we can solve using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a = 1, b = -5, and c = -36.

x = [5 ± sqrt((-5)^2 - 41(-36))] / (2*1) x = [5 ± sqrt(25 + 144)] / 2 x = [5 ± sqrt(169)] / 2 x = [5 ± 13] / 2

The two possible solutions are x = 9 and x = -4. However, since the base of a triangle cannot be negative, we discard x = -4.

So, the base of the triangle is 9 units. The height is 9 + 5 = 14 units.

Finally, we can find the area of the triangle using the formula 1/2 * base * height:

Area = 1/2 * 9 * 14 = 63 square units.