Question

if the difference between two roots of quadratic equation is 5. and the difference of their cubes is 215, find the quadratic equation. ​

Answer 1

Answer:

x² + 7x + 6 = 0 and x² - 7x + 6 = 0

Step-by-step explanation:

Given---> Difference between two roots of quadratic equation is 5 and difference of their cubes is 215.

To find---> Let roots of quadratic equation is α and β .

ATQ, Difference between roots = 5

=> α - β = 5

=> α = β + 5

ATQ, Difference between their cubes = 215

=> α³ - β³ = 215

We know that ,

a³ - b³ = ( a - b ) ( a² + b² + ab ) , applying it here , we get,

=> ( α - β ) ( α² + β² + αβ ) = 215

Putting α - β = 5 , we get,

=> 5 ( α² + β² + αβ ) = 215

=> ( α² + β² + αβ ) = 215 / 5

=> α² + β² + αβ = 43

Now putting α = β + 5 , in it we get,

=> ( β + 5 )² + β² + ( β + 5 ) β = 43

We have an identity ,

( a + b )² = a² + b² + 2ab , we get,

=> β² + 5² + 2 ( 5 ) ( β ) + β²+ β² + 5β = 43

=> β² + 25 + 10 β + β² + β² + 5β = 43

=> 3β² + 15β + 25 - 43 = 0

=> 3β² + 15β - 18 = 0

=> β² + 5β - 6 = 0

=> β² + ( 6 - 1 )β - 6 = 0

=> β² + 6β - β - 6 = 0

=> β ( β + 6 ) - 1 ( β + 6 ) = 0

=> ( β + 6 ) ( β - 1 ) = 0

If , β + 6 = 0

=> β = - 6

α = β + 5

= -6 + 5

α = - 1

We know that ,required equation is,

x² - ( α + β ) x + ( α β ) = 0

=> x² - ( -6 - 1 )x + ( - 6 ) ( -1 ) = 0

=> x² - ( -7 )x + 6 = 0

=> x² + 7x + 6 = 0

If , β - 1 = 0

=> β = 1

α = β + 5

= 1 + 5

α = 6

Required equation is

x² - ( α + β )x + αβ = 0

=> x² - ( 1 + 6 ) x + ( 1 ) ( 6 ) = 0

=> x² - ( 7 )x + 6 = 0

=> x² - 7x + 6 = 0

#Answerwithquality

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