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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 $a \neq 0$ 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.