Question

Доведіть ,що вираз -х²-18х-28 набуває від'ємного при всіх значеннях х​

Answer 1

Дальше не могу решить т к дискриминант не раскладывается

Answer 2

Proving that the Expression -x² - 18x - 28 is Always Negative

To prove that the expression -x² - 18x - 28 is always negative for all values of x, we can use the method of completing the square or analyze the quadratic equation.

Method of Completing the Square

To complete the square for the given quadratic expression -x² - 18x - 28, we can follow these steps:

1. Step 1: Find the Value of c - The value of c in the quadratic expression ax² + bx + c is given by c = (b/2)². - For the given expression -x² - 18x - 28, the value of c is calculated as c = (-18/2)² = (-9)² = 81.

2. Step 2: Rewrite the Expression - Rewrite the expression by adding and subtracting the value of c obtained in the previous step: -x² - 18x - 28 + 81 - 81.

3. Step 3: Complete the Square - Group the first three terms and factor the perfect square trinomial: -(x² + 18x + 81) - 28 - 81. - Simplify the expression: -(x + 9)² - 109.

Analysis of the Quadratic Equation

By completing the square, we have transformed the given quadratic expression into the form -(x + 9)² - 109. From this form, we can observe the following:

- The term -(x + 9)² is always non-positive or zero, as it represents a square term being multiplied by a negative value. - The term -109 is a negative constant.

Therefore, the entire expression -(x + 9)² - 109 is always negative for all real values of x.

Conclusion

Hence, we have proven that the expression -x² - 18x - 28 is always negative for all values of x, as it can be represented in the form -(x + 9)² - 109, where both terms are negative.