Question
State whether the following quadratic equations have two distinct real roots. Justify your answer.
$(x - 1)(x + 2) + 2 = 0.$
$(x - 1)(x + 2) + 2 = 0.$
✓
Answer
Main concept used:
Quadratic equation $ax^2 + bx + c = 0$ will have two distinct real roots if $D > 0 or b^2 - 4ac > 0.$
$(x - 1) (x + 2) + 2 = 0$
$\Rightarrow x^2 + 2x - x^2 + 2 = 0$
$\Rightarrow x^2 + x = 0$
$\Rightarrow x^2 + x + 0 = 0$
Now, $D = b^2 - 4ac (a = 1, b = 1, c = 0)$
$= (1)^2 - 4(1)(0) = 1$
$\Rightarrow D = 1 > 0$
So, the given equation has two distinct real roots.
Quadratic equation $ax^2 + bx + c = 0$ will have two distinct real roots if $D > 0 or b^2 - 4ac > 0.$
$(x - 1) (x + 2) + 2 = 0$
$\Rightarrow x^2 + 2x - x^2 + 2 = 0$
$\Rightarrow x^2 + x = 0$
$\Rightarrow x^2 + x + 0 = 0$
Now, $D = b^2 - 4ac (a = 1, b = 1, c = 0)$
$= (1)^2 - 4(1)(0) = 1$
$\Rightarrow D = 1 > 0$
So, the given equation has two distinct real roots.
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