Question

Докажите что выражение -у2+2у-5 при любых значениях у принимает отрицательные значения

Answer 1

-у²+2у-5=-(у²-2у+5)=-(у²-2у+4+1)=-((у-2)²+1)=-(у-2)²-1
Так как -(у-2)²<0 при любых у, то и все выражение принимает отрицательные выражения при любых у. Что и требовалось доказать.

Answer 2

Proof that the expression -у^2 + 2у - 5 takes negative values for any value of у

To prove that the expression -у^2 + 2у - 5 takes negative values for any value of у, we can analyze the quadratic equation formed by this expression. The quadratic equation is in the form of ax^2 + bx + c, where a = -1, b = 2, and c = -5.

The general form of a quadratic equation is ax^2 + bx + c = 0. To find the roots of this equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, substituting the values of a, b, and c, we get:

у = (-2 ± √(2^2 - 4(-1)(-5))) / (2(-1))

Simplifying further:

у = (-2 ± √(4 - 20)) / (-2)

у = (-2 ± √(-16)) / (-2)

Since the discriminant (b^2 - 4ac) is negative, the quadratic equation has no real roots. This means that the expression -у^2 + 2у - 5 does not intersect the x-axis and is always negative for any value of у.

Therefore, we have proven that the expression -у^2 + 2у - 5 takes negative values for any value of у.