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Location of roots

Let quadratic equation ax2 + bx + c = 0 has roots α and β. In some problems we want the roots of the equation to lie in a given interval. For this we impose conditions on a, b, and c.
 
Roots
Conditions
α, β > 0
α + β > 0, α β > 0, D ≥ 0
α, β < 0
α + β < 0, α β > 0, D ≥ 0
α < 0 < β
α β < 0
α, β > k
af(k) > 0, 62318.png, D ≥ 0
α, β < k
af(k) > 0, 62312.png, D ≥ 0
α < k < β
af(k) < 0
α, β (k1, k2)
af(k1) > 0, af(k2) > 0, 62306.png, D ≥ 0
Exactly one root says α lies in (k1k2)
f(k1) f(k2) < 0

Some important points

  1. Condition for general quadratic expression ax2 + 2hxy + by2 +2gx + 2fy + c can be factorized into two linear factors is abc + 2fghaf2bg2ch2 = 0 and h2ab > 0.
  2. If ax2 + bx + c is perfect square then b2 – 4ac = 0.
  3. If α is a repeated root of the quadratic equation f(x) = ax2 + bx + c = 0, then α is also a root of the equation f′(x) = 0.
  4. If the ratio of roots of the quadratic equation ax2 + bx + c = 0 be p : q, then pqb2 = (p + q)2 ac.




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