Applications Involving Quadratic Equations

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

Example:

A certain negative number added to the square of the number, the result is 3.75, what is the number? What is the positive number that fulfils 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, negative number is x  = - 2.5 and the positive number is x = 1.5.

Example:

A man travels 196 km by train and return in car which travel 21km/h faster. If the total journey takes 11 hours. Find the speed of train and car respectively.

Solution:
                        Let speed of train           = x km/h
                        Speed of car                   = \left( {x + 21} \right) km/h
                        Time taken by the train  = \frac{{196}}{x} hour
                        Time taken by the car     =  \frac{{196}}{{x + 21}} hour
                        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  = 28km/h
            Speed of car      = 49km/h

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