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 $$21$$km/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
$$x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}$$
$$x = \frac{{ – \left( { – 161} \right) \pm \sqrt {{{\left( { – 161} \right)}^2} – 4\left( {11} \right)\left( { – 4116} \right)} }}{{2\left( {11} \right)}}$$
$$x = \frac{{161 \pm \sqrt {207025} }}{{22}} = \frac{{161 \pm 455}}{{22}}$$
$$x = \frac{{161 + 455}}{{22}} = 28$$, $$x = \frac{{161 – 455}}{{22}} = – 13.36$$

Speed of train $$ = 28$$km/h

Speed of car    $$ = 49$$km/h