Question

Докажите,что при любых значениях a выражение -a^2-16a-65 принимает неотрицательное значение. ПОМОГИТЕ

Answer 1

Y = -a² - 16a - 65 - Это парабола с ветвями, направленными вниз, её вершина находится в точке a = - (-16)/(2*(-1))= -8.
Значение в этой точке есть максимум и он равен:
y = -(-8)² - 16*(-8) - 65 = -64 + 128 - 65 = -1
Следовательно, исходное выражение всегда отрицательно.

Answer 2

Solving the Expression for Non-Negative Values

To prove that the expression -a^2 - 16a - 65 takes non-negative values for any real number a, we can use the method of completing the square.

First, let's rewrite the expression in the form of a quadratic equation: -a^2 - 16a - 65.

Now, let's complete the square by adding and subtracting the square of half the coefficient of the linear term. The coefficient of the linear term is -16, so half of it is -8.

Adding and subtracting (-8)^2 to the expression, we get:

-a^2 - 16a - 65 = -(a^2 + 16a + 64) - 1

Now, we can rewrite the expression as a perfect square trinomial and a constant:

-(a + 8)^2 - 1

From this form, it's clear that the expression will always be less than or equal to -1, which means it will always be non-negative.

Therefore, the expression -a^2 - 16a - 65 takes non-negative values for any real number a.