Question

Решите неравенство х2 - 4х + 6 >0 ( Для записи ответа применяйте следующие данные без пробелов. Например: (-5;4) [2;3] (-оо;0]U[0;+oo) Бесконечность пишите двумя маленькими буквами оо)

Answer 1

Ответ:(-оо;+oo)

х²- 4х + 6 =0

Дискриминант =16-24=-8- отрицательный, значит, график функции, парабола ветвями вверх не пересекается с осью ох и находится выше оси ох при любом значении х. т.е. ответом служит вся числовая ось.

(-оо;+oo)

Answer 2

Solving the Inequality: x^2 - 4x + 6 > 0

To solve the inequality x^2 - 4x + 6 > 0, we can use the quadratic formula or factor the quadratic expression. Let's use the quadratic formula to find the solutions for the equation x^2 - 4x + 6 = 0, and then analyze the inequality based on those solutions.

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = -4, and c = 6. Substituting these values into the quadratic formula, we get:

x = (-(-4) ± √((-4)^2 - 4(1)(6))) / (2(1))

Simplifying further, we have:

x = (4 ± √(16 - 24)) / 2

x = (4 ± √(-8)) / 2

Since the discriminant (√(b^2 - 4ac)) is negative, the quadratic equation x^2 - 4x + 6 = 0 has no real solutions. This means that the quadratic expression x^2 - 4x + 6 is always positive or always negative.

To determine the sign of the quadratic expression, we can analyze its graph. Since the coefficient of x^2 is positive (1), the graph of the quadratic expression is a U-shaped parabola that opens upwards.

Since the quadratic expression is always positive or always negative, we need to determine the intervals where it is positive or negative. To do this, we can find the x-coordinate of the vertex of the parabola, which represents the minimum or maximum point of the graph.

The x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

In our case, a = 1 and b = -4. Substituting these values, we get:

x = -(-4) / (2(1))

x = 4 / 2

x = 2

The vertex of the parabola is located at x = 2. Since the parabola opens upwards, the quadratic expression is positive for all x values except at x = 2.

Therefore, the solution to the inequality x^2 - 4x + 6 > 0 is:

x < 2 or x > 2

In interval notation, this can be written as:

(-∞, 2) U (2, +∞)

So, the inequality x^2 - 4x + 6 > 0 is satisfied when x is less than 2 or greater than 2.

Please note that the solution is based on the analysis of the quadratic expression x^2 - 4x + 6 and its graph.