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 $a{x}^{2}+bx+c=0$ , where $a\ne 0$ and $a,b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c$ 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 , ${\left(x\right)}^{2}+{\left(x+2\right)}^{2}={\left(x+4\right)}^{2}$ .

Simplify.

${x}^{2}+{x}^{2}+4x+4={x}^{2}+8x+16$

$2{x}^{2}+4x+4={x}^{2}+8x+16$

Write in standard form.

${x}^{2}-4x-12=0$

Now factor the trinomial.

Find two numbers so that the product is $-12$ and their sum is $-4$ .

The numbers are $-6$ and $2$ .

$\left(x-6\right)\left(x+2\right)=0$

$x-6=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+2=0$

Solve each equation.

$x=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=-2$

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.