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