In the following examples, we will show how a word problem can be solved using a quadratic equation.

Recall that for a right triangle the sum of the squares of the legs is equal to the square of the Hypotenuse:

Suppose that one leg of a right triangle is 12 inches while the hypotenuse is inches.. Find the length of the other leg.

Let x be the length of the other leg. Substituting into the Pythagorean theorem we have

Since the right side is equal to This equation simplifies to

This means that x is 4 or -4.

Only 4 can be a length of the side of a triangle; so the other leg is 4 inches long.

Suppose that one leg of a right triangle is 1 more than the other leg; and the hypotenuse is 1 less than 2 times the shorter leg. Find the lengths of all the sides.

Let one side have length x. then the other side can be expressed as x + 1 (the longer leg). The hypotenuse would then be expressed as 2x - 1. So by the Pythagorean theorem we have the equation

We need to solve this equation for x. We must first expand (multiply) all of the terms:

so we have the equation

Combining like terms and setting one side to zero we have

Factoring the right hand side gives

If we set each of these factors equal to zero we get two solutions: x = 0 or x = 3. Only 3 can be the length of a side of a triangle. So the lengths are : 3, 4, and 5.

Recall that if x is an integer, then the next consecutive integer is x + 1. For example, the next consecutive integer after 7 is 7 +1 = 8.

If x is an even integer the next consecutive even integer would be x + 2. For example, the next consecutive even integer after -6 is -6 + 2 = -4.

If x is and odd integer the next consecutive odd integer would be x + 2 as well.   For example, the next consecutive odd integer after -9 is -9 + 2 = -7.

Example 3
Find two positive consecutive odd integers whose product is 99.

Let x be the first integer. Then the next odd integer is x + 2. So we have

To solve this equation first we distribute and then set one side to zero. We have

Factoring the left side gives

(x + 11)(x - 9) = 0.

So the solutions are x = -11 or x = 9. Since we are looking for positive integers, the answers are 9 and 11.

Recall the following formulas for a rectangle:

Perimeter:  P = 2L + 2 W
Area:   A = LW

The width of a rectangle is 16 feet less than 3 times the length. If the area is 35 square feet, find the dimensions of the rectangle.

Let x be the length because the width is expressed in terms of the length. So the length is 3x - 16. The total area is 35 square feet so we have the equation

To solve we first distribute:

                  35 = 3x²  - 16x

then set the left side to zero:

Factoring gives

Only the second factor will give a positive solution, so the answer is 7. The dimensions of the rectangle are: Length: 7 feet width: 5 feet.

Exercises

1.  The legs of a right triangle are inches. Find the hypotenuse.

3.  The three sides of a right triangle form three consecutive even numbers. Find the lengths of the
      three sides, measured in inches.

4.  A ladder is resting against a wall. The top of the ladder touches the wall at a height of 15 feet. Find
     the distance from the wall to the bottom of the ladder if the length of the ladder is one foot more
      than twice its distance from the wall.

6.   The width of a rectangle is 15 cm. less than 3 times the length. If the area is 42 square cm. find the
      dimensions of the rectangle.

7.  A plastic box that holds a standard audiocassette has a length 4 cm. longer than its width. The
       area of the rectangular top of the box is 77 square cm. Find the length and width of the box.

Answers