Quadratic equations are quadratic functions that are set equal to a value. A quadratic equation is an equation that can be written in the standard form ax2 + bx + c = 0, where a ≠ 0 and a , b , and c are integers.
The quadratic equations are very useful in real world situations. Here we see an example of finding the lengths of a right triangle.
The three sides of a right triangle form three consecutive even numbers. Find the lengths of the three sides, measured in feet.
First assign a variable to one side of the triangle. The smaller value is the length of the shorter leg and the higher value is the hypotenuse of the right triangle.
Let x be the length of the shorter leg. The three sides are formed by three consecutive even integers. So, x + 2 be the length of the longer leg and x + 4 be the length of the hypotenuse.
By Pythagorean Theorem, (x)2 + (x + 2)2 = (x + 4)2 .
x2 + x2 + 4x + 4 = x2 + 8 x + 16
2x2 + 4 x + 4 = x2 + 8 x + 16
Write in standard form.
x2 – 4 x – 12 = 0
Now factor the trinomial.
Find two numbers so that the product is –12 and their sum is –4.
The numbers are –6 and 2.
(x – 6)(x + 2) = 0
x – 6 = 0 or x + 2 = 0
Solve each equation.
x = 6 or x = –2
Since the length of the triangle cannot be negative, the value of x is 6. So, the length of the shorter leg is 6 ft.
The length of the longer leg is 6 + 2 or 8 ft and the hypotenuse is 6 + 4 or 10 ft.