Question

1) f(x)=x^2-2x+1 2) f(x)=x^3-2x 3) f(x)= -x^3+x^2 4) f(x)=x^2+x+1

Answer 1

x²-2x+1=0

D=(-2)²-4*1*1=0

x1=-(-2)-0/2=1

x2=-(-2)+0/2=1

Все уравнения решаются по этому способу.

Формулы:

D=b²-4ac

x=-b-√D/2a

x2=-b+√D/2a

Answer 2

To analyze the given functions, let's look at their properties and characteristics:

1. f(x) = x^2 - 2x + 1

This is a quadratic function, and it is a perfect square trinomial since the coefficient of x^2 is 1, and the coefficient of x is -2 (which is twice the coefficient of the square root term, 1).

2. f(x) = x^3 - 2x

This is a cubic function. It has odd degree and a leading coefficient of 1. The function can be factored as x(x^2 - 2), but it does not have any real roots (zeros) since the quadratic factor x^2 - 2 has no real solutions. It might have imaginary roots.

3. f(x) = -x^3 + x^2

This is also a cubic function, but the leading coefficient is -1. The function can be factored as x^2(-x + 1), so it has two real roots: x = 0 and x = 1.

4. f(x) = x^2 + x + 1

This is a quadratic function. It doesn't factor into real roots since the discriminant (b^2 - 4ac) is negative, so it has no real x-intercepts. It does have a complex root, and its graph represents a parabola that opens upwards.

To better understand these functions, it's also helpful to visualize their graphs:

1. f(x) = x^2 - 2x + 1

Graph: A parabola that opens upwards and has a vertex at (1, 0).

2. f(x) = x^3 - 2x

Graph: A cubic function with no real roots, so it doesn't cross the x-axis.

3. f(x) = -x^3 + x^2

Graph: A cubic function that intersects the x-axis at x = 0 and x = 1. It has a local maximum at (0, 0) and a local minimum at (1, 0).

4. f(x) = x^2 + x + 1

Graph: A parabola that opens upwards and does not intersect the x-axis (no real roots). It has a vertex at (-1/2, 3/4).

I hope this clarifies the characteristics of the given functions! If you have any more specific questions or need further explanations, feel free to ask.