Transcription of Further Pure 1 - Mathsbox
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Further Pure 1 - Mathsbox
Further pure 1 Summary Notes1. Roots of quadratic EquationsFor a quadratic equationax2 + bx + c = 0 with roots and Sum of the roots Product of rootsab =caa +b = ba If the coefficients a,b and c are real then either and are real or and arecomplex conjugatesOnce the value + and have been found, new quadratic equations can be formedwith roots :Roots 2 and 2 Sum of roots(a +b )2 2abProduct of roots(ab )2 3 and 3 Sum of roots(a +b )3 3ab (a +b )Product of roots(ab )31aand1bSum of rootsa +babProduct of roots1abThe new equation becomesx2 (sum of new roots)x + (product of new roots) = 0 The questions often ask forinteger coefficients!Don t forget the = 0 ExampleThe roots of the quadratic equation3x are and .Determine a quadratic equation with integer coefficients which has roots2+ 4x 1 = 0a3b and ab 3a +b = 43ab = 13 Step 1 :Step 2 :Sum of new rootsa3b + ab3 = ab (a2 + b 2)= 1 ((a +b)23 2ab )= 13 169+2 3 = 3 : Product of rootsa3b ab 3 = a4b4 = (ab )4 =181 Step 4 : Form the new equationx2 +22x +12781= 081x2 + 66x + 1 = 02.
Further Pure 1 Summary Notes 1. Roots of Quadratic Equations For a quadratic equation ax2 + bx + c = 0 with roots α and β Sum of the roots Product of roots ab = c a a + b = – b a
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