Lodovico Ferrari’s Method

 

Ferrari was the first mathematician to solve the quartic, and he did so in a very sophisticated manner. For supplementary information on Ferrari and each proceeding mathematician see Appendix I.
The first method adopted by Ferrari looks at an equation of the form (2) (following [1], Chapter 7.4 & [2], Chapter 6)
This can be rewritten by dividing the equation through by A so that the coefficient of the x^4 term is reduced to one. Further substituting x=X-B/4A gets rid of the x^3 term which can be seen below as follows. x^4+B/A x^3+C/A x^2+D/A x+E/A=0(X-B/4A)^4+B/A (X-B/4A)^3+C/A (X-B/4A)^2+D/A (X-B/4A)+E/A=0(3)

Expanding brackets and simplifying terms we have the following
D/A (X-B/4A)=DX/A-BD/(4A^2 ) C/A (X-B/4A)^2=C/A [(X-B/4A)(X-B/4A)] C/A [X^2-BX/4A-BX/4A+B^2/(16A^2 )] C/A [X^2-2BX/4A+B^2/(16A^2 )] C/A [X^2-BX/2A+B^2/(16A^2 )] (CX^2)/A-BCX/(2A^2 )+(B^2 C)/(16A^3 ) B/A (X-B/4A)^3=B/A [(X-B/4A)^2 (X-B/4A)] B/A [(X^2-BX/2A+B^2/(16A^2 ))(X-B/4A)] B/A [(X^3-(BX^2)/2A+(B^2 X)/(16A^2 )-(BX^2)/4A+(B^2 X)/(8A^2 )-B^3/(64A^3 ))] B/A [(X^3-(2BX^2)/4A-(BX^2)/4A+(B^2 X)/(16A^2 )+(〖2B〗^2 X)/(16A^2 )-B^3/(64A^3 ))] B/A [(X^3-(3BX^2)/4A+(3B^2 X)/(16A^2 )-B^3/(64A^3 ))] (BX^3)/A-(3B^2 X^2)/(4A^2 )+(3B^3 X)/(16A^3 )-B^4/(64A^4 ) (X-B/4A)^4=(X-B/4A)^2 (X-B/4A)^2 (X^2-BX/2A+B^2/(16A^2 ))(X^2-BX/2A+B^2/(16A^2 )) X^4-(BX^3)/2A+(B^2 X^2)/(16A^2 )-(BX^3)/2A+(B^2 X^2)/(4A^2 )-(B^3 X)/(32A^3 )+(B^2 X^2)/(16A^2 )-(B^3 X)/(32A^3 )+B^4/(256A^4 ) X^4-(2BX^3)/2A+(〖2B〗^2 X^2)/(16A^2 )+(B^2 X^2)/(4A^2 )-(〖2B〗^3 X)/(32A^3 )+B^4/(256A^4 ) X^4-(BX^3)/A+(B^2 X^2)/(8A^2 )+(〖2B〗^2 X^2)/(8A^2 )-(B^3 X)/(16A^3 )+B^4/(256A^4 ) X^4-(BX^3)/A+(〖3B〗^2 X^2)/(8A^2 )-(B^3 X)/(16A^3 )+B^4/(256A^4 )

Combining each of these terms we have that (3) equates to
X^4+((3B^2)/(8A^2 )-(3B^2)/(4A^2 )+C/A) X^2+((2B^3)/(16A^3 )-BC/(2A^2 )+D/A)X+(B^4/(256A^4 )-B^4/(64A^4 )+(B^2 C)/(16A^3 )-BD/(4A^2 )+E/A)=X^4+((8AC-3B^2)/(8A^2 )) X^2+((B^3-4ABC+8A^2 D)/(8A^3 ))X+((-3B^4+16AB^2 C-64A^2 BD+256A^3 E)/(256A^4 ))=0(4)Thus in general we have a reduced quartic of the form
X^4+aX^2+bX+c=0(5)
It follows
X^4+aX^2=-bX-c(6)
Now Ferrari added the term aX^2+a^2 to both sides of the equation, so we are able to complete the square as follows.
X^4+aX^2+aX^2+a^2=-bX-c+(aX^2+a^2 ) X^4+2a^2+a^2=(aX^2+a^2 )-bX-c(X^2+a)^2-a^2+a^2=aX^2+a^2-bX-c(X^2+a)^2=aX^2+a^2-bX-c
Additionally by the introduction of 2(X^2+a)y+y^2 we can rewrite this further like so
(X^2+a)^2+2(X^2+a)y+y^2=aX^2+a^2-bX-c+2(X^2+a)y+y^2 ((X^2+a)+y)^2

 

 

 

Sample Solution

orn a blank canvas and their lives are determined by varying experiences, positive interactions whilst acquiring different attributes and not due to DNA. She believed men and women were born equal and brains were malleable by culture, life experiences and desires, whereas Freud (1961) argued this and believed men and women’s brains were very different and the way they were formed was wholly down to DNA. Freud (1961) founded the study of psychoanalysis, he believed traumas and unresolved issues during childhood could be coaxed out by sensitive talk therapy. This can be seen in settings today by way of one:one counselling or through emotional literacy. Place2Be (1994) delivers school based therapeutic interventions. A large study was carried out on 47,000 children in primary schools, based on the findings of Place2Be, children’s social and emotional wellbeing had improved as perceived by their family members and teachers following interventions (White Et al., 2009). Supporting this Fox and Butler (2007) carried out a small-scale study on 219 pupils, attempting to discover whether counselling is successful in schools. The results suggested that the treatment delivered to the children, had less distress and fewer problems were observed. However, they recommended further research is required, to discover when and to who it was more of a success to. The evidence suggests that pupils receiving psychological therapies were a success however, The National Institute for Health and Care Excellence (NICE, 2010) expressed the opinion and recommended several forms of therapy as first-line interventions, not talk therapy or counselling in isolation. Another essential point regarding one:one counselling and emotional literacy is the relationship a child has with the facilitator. Mindmatters (n.d.) advocates how vital it is for children to have a trusting and caring relationship with the individual, as a result of this trust and understanding, will promote open communication. Similarly, Bowlby (1969) suggested that children come into this world with an innate response of survival; form attachments to survive. Consequently, if secure attachments are not formed this would lead to distress and negative experiences concluding that a trusting relationship is not formed between the pupil and the facilitator. Belsky and Fearon (2002) support Bowlby and state ‘poor attachment outcomes are associated with long-term adverse consequences in cognitive, adaptive, and behavioral domains’. Comparing Maslow (1968c), Bronfenbrenner (1979) Sameroff (2010) and Shonokoff an

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