Example 7.82 from Elementary Algebra, 7.6 Quadratic Equations

Quadratic Equations: example 7.82

Justine wants to put a deck in the corner of her backyard in the shape of a right triangle, as shown below. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the deck.

This figure is a right triangle. The vertical leg is labeled “x – 7”. the horizontal leg, the base, is labeled “x”. The hypotenuse is labeled “17”.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

A boat’s sail is a right triangle. The length of one side of the sail is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the sail.

How did it go?

Step 1. Read the problem.
Step 2. Identify what you are looking for.	We are looking for the lengths of the sides of the deck.
Step 3. Name what you are looking for. One side is 7 less than the other.	Let x = length of a side of the deck x − 7 = length of other side
Step 4. Translate into an equation. Since this is a right triangle we can use the Pythagorean Theorem.	$a^{2} + b^{2} = c^{2}$
Substitute in the variables.	$x^{2} + {(x - 7)}^{2} = 17^{2}$
Step 5. Solve the equation.	$x^{2} + x^{2} - 14 x + 49 = 289$
Simplify.	$2 x^{2} - 14 x + 49 = 289$
It is a quadratic equation, so get zero on one side.	$2 x^{2} - 14 x - 240 = 0$
Factor the greatest common factor.	$2 (x^{2} - 7 x - 120) = 0$
Factor the trinomial.	$2 (x - 15) (x + 8) = 0$
Use the Zero Product Property.	$2 \neq 0$	$x - 15 = 0$	$x + 8 = 0$
Solve.	$2 \neq 0$	$x = 15$	$x = −8$
Since x is a side of the triangle, $x = −8$ does not make sense.	$2 \neq 0$	$x = 15$	$x = −8$
Find the length of the other side.
If the length of one side is
then the length of the other side is

	8 is the length of the other side.
Step 6. Check the answer. Do these numbers make sense?

Step 7. Answer the question.	The sides of the deck are 8, 15, and 17 feet.

Step 1. Read the problem.
Step 2. Identify what you are looking for.	We are looking for the lengths of the sides of the deck.
Step 3. Name what you are looking for. One side is 7 less than the other.	Let x = length of a side of the deck x − 7 = length of other side
Step 4. Translate into an equation. Since this is a right triangle we can use the Pythagorean Theorem.	$a^{2} + b^{2} = c^{2}$
Substitute in the variables.	$x^{2} + {(x - 7)}^{2} = 17^{2}$
Step 5. Solve the equation.	$x^{2} + x^{2} - 14 x + 49 = 289$
Simplify.	$2 x^{2} - 14 x + 49 = 289$
It is a quadratic equation, so get zero on one side.	$2 x^{2} - 14 x - 240 = 0$
Factor the greatest common factor.	$2 (x^{2} - 7 x - 120) = 0$
Factor the trinomial.	$2 (x - 15) (x + 8) = 0$
Use the Zero Product Property.	$2 \neq 0$	$x - 15 = 0$	$x + 8 = 0$
Solve.	$2 \neq 0$	$x = 15$	$x = −8$
Since x is a side of the triangle, $x = −8$ does not make sense.	$2 \neq 0$	$x = 15$	$x = −8$
Find the length of the other side.
If the length of one side is
then the length of the other side is

	8 is the length of the other side.
Step 6. Check the answer. Do these numbers make sense?

Step 7. Answer the question.	The sides of the deck are 8, 15, and 17 feet.

Quadratic Equations: example 7.82

Solution

Your turn

Quadratic Equations: example 7.82

Solution

Your turn