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Relation Between Roots of the Equation and Coefficient of the Terms in the Equation Equations Reducible to Quadratic Form

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a and b are the roots of the equation ax2 + bx + c = 0 then,

(1)`alpha+beta=(-b+sqrt(b^2-4ac))/(2a)+(-b-sqrt(b^2-4ac))/(2a)`

  `=(-b+sqrt(b^2-4ac)-b-sqrt(b^2-4ac))/(2a)`

  `=-(2b)/(2a)`

  `therefore alpha+beta=-b/a`

(2) `alphaxxbeta=(-b+sqrt(b^2-4ac))/(2a)xx(-b-sqrt(b^2-4ac))/(2a)`

`=((-b+sqrt(b^2-4ac))xx(-b-sqrt(b^2-4ac)))/(4a^2)`

`=(b^2-(b^2-4ac))/(4a^2)`

`=(4ac)/(4a^2)`

`=c/a`

`therefore alpha beta=c/a`

  Ex. (1) If a and b are the roots of the quadratic equation 2x2 + 6x - 5 = 0, then find (a + b) and a × b.
Solution : Comparing 2x2 + 6x - 5 = 0 with ax2 + bx + c = 0.
∴a = 2, b = 6 , c = -5
∴a + b = -b/a = -6/2 = -3
and a × b =c/a =−5/2

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