Question

Докажите, что при любом значении m разность выражений 9+8m и (m+4)^2 принимает отрицательные значения

Answer 1

9 + 8m - (m + 4)² = 9 + 8m - m² - 8m - 16 = -m² - 7

Answer 2

Proof that the expression 9 + 8m - (m + 4)^2 takes negative values for any value of m

To prove that the expression 9 + 8m - (m + 4)^2 takes negative values for any value of m, we can simplify the expression and analyze its behavior.

Let's start by expanding the square term (m + 4)^2:

(m + 4)^2 = m^2 + 8m + 16

Now, substitute this expression back into the original expression:

9 + 8m - (m + 4)^2 = 9 + 8m - (m^2 + 8m + 16)

Simplifying further:

9 + 8m - (m^2 + 8m + 16) = 9 + 8m - m^2 - 8m - 16

Combining like terms:

9 + 8m - m^2 - 8m - 16 = -m^2 + (8m - 8m) + (9 - 16)

Simplifying:

-m^2 + (8m - 8m) + (9 - 16) = -m^2 - 7

From this simplified expression, we can see that the coefficient of the squared term is negative (-1), which means the parabola opens downwards. Therefore, the expression -m^2 - 7 takes negative values for any value of m.

Hence, we have proven that the expression 9 + 8m - (m + 4)^2 takes negative values for any value of m.