Applications Involving Quadratic Equations

Quadratic equations have many applications in the arts and sciences, business, economics, medicine and engineering.

Example:

A certain negative number is added to the square of the number, and the result is 3.75. What is the number? What is the positive number that fulfills this condition?

Solution:

Let x be a negative number.

By the given condition
x + {x^2} = 3.75
{x^2} + x - 3.75 = 0

Let a = 1, b = 1 and c = - 3.75

Using the quadratic formula, we have
x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
x = \frac{{ - 1 \pm \sqrt {{1^2} - 4\left( 1 \right)\left( { - 3.75} \right)} }}{{2\left( 1 \right)}}
x = \frac{{ - 1 \pm \sqrt {16} }}{2} = \frac{{ - 1 \pm 4}}{2}
x = \frac{{ - 1 - 4}}{2}         ,         x = \frac{{ - 1 + 4}}{2}
x = \frac{{ - 5}}{2} = - 2.5    ,         x = \frac{3}{2} = 1.5

Hence, the negative number is x = - 2.5 and the positive number is x = 1.5.

 

Example:

A man travels 196 km by train and returns in a car which travels 21km/h faster. If the total journey takes 11 hours, find the speed of the train and car respectively.

Solution:
Let the speed of the train  = x km/h.
Speed of car  = \left( {x + 21} \right) km/h
Time taken by the train  = \frac{{196}}{x} hours
Time taken by the car  = \frac{{196}}{{x + 21}} hours
Total time  = 11 hours

Then by the condition
\frac{{196}}{x} + \frac{{196}}{{x + 21}} = 11
\frac{{196\left( {x + 21} \right) + 196x}}{{x\left( {x + 21} \right)}} = 11
196x + 4116 + 196x = 11{x^2} + 231x
11{x^2} - 161x - 4116 = 0

Let a = 11, b = - 161 and c = - 4116.

Using the quadratic formula, we have
x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
x = \frac{{ - \left( { - 161} \right) \pm \sqrt {{{\left( { - 161} \right)}^2} - 4\left( {11} \right)\left( { - 4116} \right)} }}{{2\left( {11} \right)}}
x = \frac{{161 \pm \sqrt {207025} }}{{22}} = \frac{{161 \pm 455}}{{22}}
x = \frac{{161 + 455}}{{22}} = 28x = \frac{{161 - 455}}{{22}} = - 13.36

Speed of train  = 28km/h

Speed of car     = 49km/h