Edexcel A-Level Mathematics AS Fp1 2 3 Forming Quadratic Equations with 1 1 1 1 2 QuestionsPractise forming quadratic equations whose roots are algebraic transformations of α and β, using sums and products only.SyllabusFirst assessment 2019CourseMathematics YMA01LevelAS
Exam pointsclear fractions to give the new quadratic in integer-coefficient formuse earlier α²+β² or α³+β³ results when forming the new equation
Edexcel A-Level Mathematics AS Fp1 2 3 Forming Quadratic Equations with 1 1 1 1 2 Questions question 1[Maximum number: 6]The quadratic equation2x2−3x+7=02 x^{2}-3 x+7=02x2−3x+7=0has roots α\alphaα and β\betaβWithout solving the equation,find a quadratic equation which has roots(α−1β2) and (β−1α2)\left(\alpha-\frac{1}{\beta^{2}}\right) \text { and }\left(\beta-\frac{1}{\alpha^{2}}\right)(α−β21) and (β−α21)giving your answer in the form px2+qx+r=0p x^{2}+q x+r=0px2+qx+r=0 where p, q and r are integers to be determined.Show AnswerFor roots α−1β2\alpha-\frac1{\beta^2}α−β21 and β−1α2\beta-\frac1{\alpha^2}β−α21:Sum=α+β−α2+β2α2β2=32−−19/4(7/2)2=18598.\text{Sum}=\alpha+\beta-\frac{\alpha^2+\beta^2}{\alpha^2\beta^2} =\frac32-\frac{-19/4}{(7/2)^2}=\frac{185}{98}.Sum=α+β−α2β2α2+β2=23−(7/2)2−19/4=98185.Product=αβ−α+βαβ+1α2β2=72−37+449=30998.\text{Product}=\alpha\beta-\frac{\alpha+\beta}{\alpha\beta}+\frac1{\alpha^2\beta^2} =\frac72-\frac37+\frac4{49}=\frac{309}{98}.Product=αβ−αβα+β+α2β21=27−73+494=98309.x2−18598x+30998=0.x^2-\frac{185}{98}x+\frac{309}{98}=0.x2−98185x+98309=0.98x2−185x+309=0.98x^2-185x+309=0.98x2−185x+309=0.Add to Test