# The height of the triangle is 3 cm less than its base. find the base of the triangle if its area is 14 cm2.

The area of a triangle is calculated by the formula:

S = 1/2 * a * h,

where S is the area of the triangle,

a – the base of the triangle,

h is the height of the triangle.

Let x be the base of the triangle, then (x -3) is the height of the triangle.

Let’s substitute the values into the formula:

14 = 1/2 * x * (x – 3).

14 = 1/2 * x ^ 2 – 3/2 * x – to simplify the solution, multiply both sides of the equation by the number 2.

14 * 2 = 2 * 1/2 * x ^ 2 – 2 * 3/2 * x.

28 = x ^ 2 – 3 * x.

x ^ 2 – 3 * x – 28 = 0.

Let’s solve the quadratic equation.

x ^ 2 – 3 * x – 28 = 0

Find the discriminant of the quadratic equation:

D = b ^ 2 – 4 * a * c = (-3) ^ 2 – 4 * 1 * (-28) = 9 + 112 = 121.

Since the discriminant is greater than zero, the quadratic equation has two real roots. However, taking into account that the base of the triangle cannot have a negative value, the root of the equation will be one.

x = (3 + √121) / (2 * 1) = (3 + 11) / 2 = 14/2 = 7 cm.

The base of the triangle is 7 cm.