Question

F(x)=4x^2-8x+1 подробное решение

Answer 1

Ответ:

Пошаговое объяснение:

4x^2-8x+1=0

D=8^2-4*4*1=16-16=0, D=0 следовательно один корень x1=x2=x

x=-b/2a=-(-8)/2*4=8/8=1

f(x)=1

Answer 2

To find the detailed solution for the function $F(x) = 4x^2 - 8x + 1$, we need to understand the following steps:

1. Evaluate the function for specific values of $x$.
2. Find the x-intercepts (roots) of the function.
3. Determine the vertex of the parabola (minimum or maximum point).
4. Sketch the graph of the function.

Let's start with step by step explanation:

1. Evaluate the function for specific values of $x$: We can plug in any value of $x$ to find the corresponding value of $F(x)$. For example, if we want to find $F(0)$: $F(0) = 4(0)^2 - 8(0) + 1 = 1$ Similarly, if we want to find $F(2)$: $F(2) = 4(2)^2 - 8(2) + 1 = 4(4) - 16 + 1 = 16 - 16 + 1 = 1$

2. Find the x-intercepts (roots) of the function: The x-intercepts are the points where the graph of the function intersects the x-axis (i.e., $F(x) = 0$). To find the x-intercepts, we need to solve the quadratic equation $F(x) = 4x^2 - 8x + 1 = 0$.

   The quadratic equation $ax^2 + bx + c = 0$ can be solved using the quadratic formula: $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$

   For $F(x) = 4x^2 - 8x + 1$, the coefficients are $a = 4$, $b = -8$, and $c = 1$. Plugging these values into the quadratic formula, we get: $x = \frac{{-(-8) \pm \sqrt{{(-8)^2 - 4 \cdot 4 \cdot 1}}}}{{2 \cdot 4}}$ $x = \frac{{8 \pm \sqrt{{64 - 16}}}}{8}$ $x = \frac{{8 \pm \sqrt{48}}}{8}$

   Now, we can simplify the expression inside the square root: $\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}$

   Therefore, the x-intercepts are: $x = \frac{{8 + 4\sqrt{3}}}{8}$ and $x = \frac{{8 - 4\sqrt{3}}}{8}$

3. Determine the vertex of the parabola (minimum or maximum point): The vertex of the parabola represented by $F(x) = 4x^2 - 8x + 1$ can be found using the formula $x = -\frac{b}{2a}$.

   For $F(x) = 4x^2 - 8x + 1$, the coefficients are $a = 4$ and $b = -8$. $x = -\frac{-8}{2 \cdot 4} = -\frac{-8}{8} = 1$

   To find the corresponding value of $F(x)$ at $x = 1$: $F(1) = 4(1)^2 - 8(1) + 1 = 4 - 8 + 1 = -3$

   Therefore, the vertex of the parabola is at the point (1, -3).

4. Sketch the graph of the function: We have enough information to sketch the graph of the function $F(x) = 4x^2 - 8x + 1$