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Question
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
`x^2-4x-1=0`
Solution
Given:
`x^2-4x-1=0`
On comparing it with `ax^2+bx+c=0`
a =1,b = -4 and c = -1
Discriminant D is given by:
D=(b^2-4ac)
=`(-4)^2-4xx1xx(-1)`
=`16+4`
=20
=`20>0`
Hence, the roots of the equation are real.
Roots α and β are given by:
`α =(-b+sqrt(D))/(2a)=(-(-4)+sqrt(20))/(2xx1)=(4+2sqrt(5))/2=(2(2+sqrt(5)))/2=(2+sqrt(5))`
`β=(-b-sqrt(D))/(2a)=(-(-4)-sqrt(20))/2=(4-2sqrt(5))/2=(2(2-sqrt(5)))/2=(2-sqrt5)`
Thus, the roots of the equation are `(2+sqrt5) and (2-sqrt5)`
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