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Question

For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorization.

(i)   83,43  (ii) 218,516  (iii) 23,9    (iv) 325,12


Solution

A quadratic equation can be written as
x squared minus left parenthesis s u m space o f space r o o t s right parenthesis x plus p r o d u c t space o f space r o o t s equals 0

i)
  equals x squared minus fraction numerator negative 8 over denominator 3 end fraction x plus 4 over 3 space equals 0 equals space space x squared plus 8 over 3 x plus 4 over 3 equals 0 equals 3 x squared plus 8 x plus 4 equals 0 equals 3 x squared plus 6 x plus 2 x plus 4 equals 0 equals 3 x left parenthesis x plus 2 right parenthesis plus 2 left parenthesis x plus 2 right parenthesis equals 0 equals left parenthesis 3 x plus 2 right parenthesis left parenthesis x plus 2 right parenthesis equals 0 x equals fraction numerator negative 2 over denominator 3 end fraction comma negative 2
ii)
equals x squared minus 21 over 8 x plus 5 over 16 space equals 0 equals 16 x squared minus 42 x plus 5 equals 0 equals 16 x squared minus 40 x minus 2 x plus 5 equals 0 equals 8 x left parenthesis 2 x minus 5 right parenthesis minus left parenthesis 2 x minus 5 right parenthesis equals 0 equals left parenthesis 8 x minus 1 right parenthesis left parenthesis 2 x minus 5 right parenthesis equals 0 x equals 1 over 8 comma 5 over 2

iii)
equals x squared minus left parenthesis negative 2 square root of 3 right parenthesis x minus 9 space equals 0 equals x squared plus 2 square root of 3 x minus 9 equals 0 equals x squared plus 3 square root of 3 x minus square root of 3 x minus 9 equals 0 equals x left parenthesis x plus 3 square root of 3 right parenthesis minus square root of 3 left parenthesis x plus 3 square root of 3 right parenthesis equals 0 equals left parenthesis x minus square root of 3 right parenthesis left parenthesis x plus 3 square root of 3 right parenthesis equals 0 x equals square root of 3 comma negative 3 square root of 3

iv)
equals x squared minus fraction numerator negative 3 over denominator 2 square root of 5 end fraction x minus 1 half space equals 0 equals 2 square root of 5 x squared plus 3 x minus square root of 5 equals 0 equals 2 square root of 5 x squared plus 5 x minus 2 x minus square root of 5 equals 0 equals square root of 5 x left parenthesis 2 x plus square root of 5 right parenthesis minus left parenthesis 2 x plus square root of 5 right parenthesis equals 0 equals left parenthesis square root of 5 x minus 1 right parenthesis left parenthesis 2 x plus square root of 5 right parenthesis equals 0 x equals fraction numerator 1 over denominator square root of 5 end fraction comma fraction numerator negative square root of 5 over denominator 2 end fraction
 

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