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Word Problems: Quadratic Equations

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 a x 2 + b x + 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.

Example :

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 .

Simplify.

x 2 + x 2 + 4 x + 4 = x 2 + 8 x + 16

Combine like terms .

2 x 2 + 4 x + 4 = x 2 + 8 x + 16

Write in standard form.

x 2 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

Use zero product property .

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.

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