Question

The equation 2.x2 + 2(p+1)x+p=0, where p is real, always has roots that are (a) Equal (b) Equal in magnitude but opposite in sign (c) Irrational (d) Real

JEE/Engineering Exams

Maths

Solution

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The discriminant of a quadratic equation ( a x^{2}+b x+c=0 ) is given by ( b^{2}-4 a c ) ( a=2, b=2(p+1) ) and ( c=p ) ( b^{2}-4 a c=[2(p+1)]^{2}-4(2 p)=4(p+1)^{2}-8 p ) ( =4left[(p+1)^{2}-2 pright]=4left[left(p^{2}+2 p+1right)-2 pright] ) ( =4left(p^{2}+1right) ) For any real value of ( p, 4left(p^{2}+1right) ) will always be positiv as ( p^{2} ) cannot be negative for real ( p ) Hence, the discriminant ( b^{2}-4 a c ) will always be positiv When the discriminant is greater than ( ^{circ} 0 ) ' or is postitive then the roots of a quadratic equation will be real.