State whether the quadratic equation $(x-1)(x+2)+2=0$ has two distinct real roots. Justify your answer.
(A) Given equation is $(x-1)(x+2)+2=0$.
Expanding the terms,we get $x^{2} + 2x - x - 2 + 2 = 0$.
Simplifying,we get $x^{2} + x = 0$.
Comparing this with the standard form $ax^{2} + bx + c = 0$,we have $a = 1, b = 1, c = 0$.
The discriminant $D$ is given by $D = b^{2} - 4ac$.
Substituting the values,$D = (1)^{2} - 4(1)(0) = 1 - 0 = 1$.
Since $D > 0$,the quadratic equation has two distinct real roots.
Expanding the terms,we get $x^{2} + 2x - x - 2 + 2 = 0$.
Simplifying,we get $x^{2} + x = 0$.
Comparing this with the standard form $ax^{2} + bx + c = 0$,we have $a = 1, b = 1, c = 0$.
The discriminant $D$ is given by $D = b^{2} - 4ac$.
Substituting the values,$D = (1)^{2} - 4(1)(0) = 1 - 0 = 1$.
Since $D > 0$,the quadratic equation has two distinct real roots.
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