Check out these four great word problems involving quadratic equations in this lesson.

**Problem #1: **A rectangular garden has an area of 14 m^{2} and a perimeter of 18 meters. Find the dimensions of the rectangular garden. The figure below shows how to set up the problem.

**Solution:**

Let w = width and l = length

2l + 2w = 18

lw = 14

Divide both sides of 2l + 2w = 18 by 2 to get l + w = 9.

Using l + w = 9, solve for l. We get l = 9 - w.

Substitute 9 - w for l in lw = 14

( 9 - w)w = 14

9w - w^{2} = 14

9w - w^{2} - 14 = 0

- w^{2} + 9w - 14 = 0

w^{2} - 9w + 14 = 0

(w - 7)(w - 2) = 0

w = 7 and w = 2

**Problem #2: **The sum of two numbers is 12 and their product is 35. What are the two numbers?

**Solution: **

Let n and m be the two numbers.

n + m = 12 (1)

n × m = 35 (2)

Using (1), n = 12 - m

(12 - m) × m = 35

12m - m- m

m

(m - 5)(m - 7) = 0

m = 5 and m = 7. The two numbers are 5 and 7

**Problem #3: **The quadratic equation for the cost in dollars of producing automobile tires is given below where x is the number of tires the company produces. Find the number of tires that will minimize the cost.

**Solution: **The standard form of a quadratic equation is ax² + bx + c. To solve this problem, we just need 2 important concepts about quadratic equations. First, when we are trying to maximize or minimize, we need to use the formula below that will help us find the x-coordinate of the vertex. Second, if a > 0, the vertex is a minimum. if a < 0, the vertex is a maximum.

x =

-b
2a

Since a = 0.00002 and 0.00002 is bigger than 0, the quadratic equation will give a minimum.

x =

- -0.04
2 × 0.00002

x =

0.04
0.00004

=1000
To minimize the cost, the company should produce 1000 tires.

**Problem #4: **You want to frame a collage of pictures with a 9-ft strip of wood. What dimensions will help you maximize the area?

Area = l × w Perimeter = 2l + 2w

9 = 2l + 2w. Solve for l and replace l in the formula for the area.

9 - 2w = 2l

l =

9 - 2w
2

A =

9 - 2w
2

× w
A =

9w - 2w^{2}
2

A = -w^{2} + 4.5w

A = -w^{2} + 4.5w + 0

Since a = -1 and -1 is smaller than 0, the quadratic equation will give a maximum.

x =

-b
2a

w =

-4.5
-2

= 2.25
l =

9 - 2 × 2.25
2

l =

9 - 4.5
2

= 2.25
To maximize the area, l = w = 2.25

If you found the word problems involving quadratic equations on this lesson difficult to understand, review the lesson about factoring trinomials