the concern is discover the selection of worths of $c$ because that which the expression $4x^2-4x+4c^2-8$ is non-negative for all real values the $x$.

You are watching: If the discriminant is less than zero

I got the discriminant but I don"t know why it has to be less than zero. If it is non-negative, shouldn"t the be optimistic and much more than zero?

Can someone aid me explain this, thanks :)

Here that is basic to complete the square come obtain

$$(2x-1)^2+4c^2-9\ge 0$$

You should recognise her discriminant here. The minimum value of the left-hand side clearly occurs v $2x=1$ and you should have the ability to finish native here.

With $ax^2+bx+c$ girlfriend can finish the square cleanly by multiplying by $4a$ to acquire $$4a^2x^2+4abx+4ac=(2ax+b)^2+4ac-b^2$$

You simply need to bear in mind that if $a$ is negative, multiplying by $4a$ changes the sign.

The kind of the equation in this inquiry meant that no fiddling to be necessary. However the general case shows you where the discriminant comes in, and also might help you to understand the sign.

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edited Aug 10 "15 at 9:40
answered Aug 10 "15 in ~ 9:33

note BennetMark Bennet
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Do not confuse the authorize of the discriminant v the authorize of the expression. A negative discriminant ensures that the expression has no actual root, i.e. The expression keeps the very same sign for every $x$.

The negative discriminant is not enough: the expression should be optimistic for at least one value of $x$, then it is hopeful for all. (It suffices the the leading coefficient be positive, which is the instance here.)

$$\forall x:ax^2+bx+c>0\iff b^2-4ac0.$$

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edited Aug 10 "15 in ~ 9:47
reply Aug 10 "15 in ~ 9:40
user65203user65203
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Another possible approach : consider the duty $$f(x)=4x^2-4x+4c^2-8$$ $$f"(x)=8x-4$$ The derivative cancels for $x=\frac 12$ (which coincides to a minimum) and, in ~ this point, the worth of the duty is $$f(\frac 12)=4c^2-9$$ and you want this to constantly be non negative.

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answered Aug 10 "15 in ~ 9:48

Claude LeiboviciClaude Leibovici
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The Parabula is "smiling" and also hence in order it will certainly be positive, you require that there will be no intersection through the $x$ axis, that is there room no remedies to the equation$$4x^2-4x+4c^2-8=0.$$That happens if and only if the discriminant is negative.

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answer Aug 10 "15 in ~ 9:33

Ofir SchnabelOfir Schnabel
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