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Quadratic Equations with Real Coefficients

Consider the quadratic equation
ax2 + bx + c = 0   ...(1)
 
where a, b, c R and a ≠ 0.
 
Roots of the equation are given by
x = 62595.png
 
Now, if we look at these roots, we observe that the roots depend upon the value of the quantity b2 – 4ac.
 
This quantity is generally denoted by D and is known as the discriminant of the quadratic equation (1).
 
We also observe the following results:
62589.png

 

Notes:
  • If abc ∈ Q and b2 – 4ac is positive but not a perfect square, then roots are irrational and they always occur in conjugate pair like 2 + 64865.png and 2 – 64860.png. However, if abc are irrational numbers and b2 – 4ac is positive but not a perfect square, then the roots may not occur in conjugate pairs. For example, the roots of the equation x2 – (5 + 64856.pngx + 564852.png= 0 are 5 and 64848.png which do not form a conjugate pair.
  • If b2 – 4ac < 0, then roots of equations are complex. If ab, and c are real then complex roots occur in conjugate pair like of the form p + iq and p – iq. If all coefficients are not real then complex roots are not conjugate.
  • Relations between roots and coefficients
     
    α + β = 63161.png = 63155.png
     
    and α β = 63149.png = 63143.png
     
    Also, if sum of roots is S and product is P, then quadratic equation is given by x2 – Sx + P = 0.
  • Symmetric functions of roots
    • α2 + β2 = (α + β)2 – 2αβ
    • (α – β)2 = (α + β)2 – 4αβ
    • α3 + β3 = (α + β)3 – 3αβ(α + β)
    • αn + βn = (α + β)(αn–1 + βn–1)
       
      – αβ(αn–2 + βn–2)
       
      or Sn = (α + β)Sn–1 – αβSn–2
       
      where Sn = αn + βn




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