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3. Determine the nature of roots of the following quadratic equations.

(1) \(x^2\) – 4\(x\) + 4 = 0.

(2) 2\(y^2\) – 7y + 2 = 0

(3) \(m^2\) + 2m + 9 = 0

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(1)   \(x^2\) – 4\(x\) + 4 = 0.

Sol.  \(x^2\) – 4\(x\) + 4 = 0.  comparing with  a\(x^2\) + b\(x\) + c = 0
        We get, a = 1 b = –4 c = 4
∴      \(b^2\) – 4ac = \((-4)^2\) – 4 × 1 × 4
∴               Δ    = 16 – 16
∴               Δ    =  0
∴      \(b^2\) – 4ac = 0
∴   The roots of the equation are real and equal.

 

(2)   2\(y^2\) – 7y + 2 = 0

Sol. 2\(y^2\) – 7y + 2 = 0  comparing with  a\(x^2\) + b\(x\) + c = 0
       We get, a = 2, b = –7 and c = 2
       \(b^2\) – 4ac = \((-7)^2\) – 4 × 2 × 2
∴              Δ   =  49 – 16
∴              Δ   =  33
∴     \(b^2\) – 4ac > 0
The roots of equation are real and unequal.

 

(3)   \(m^2\) + 2m + 9 = 0

Sol. \(m^2\) + 2m + 9 = 0 comparing with  a\(x^2\) + b\(x\) + c = 0
        a = 1 b = 2 and c = 9
        \(b^2\) – 4ac = \(2^2\) – 4 × 1 × 9
∴               Δ   = 4 – 36
∴               Δ   = –32
∴     b2 – 4ac < 0
∴     The roots of equation are not real.

uÉaÉïxÉqÉÏMüUhÉ August 10 , 2018 0 Comments 42 views