Question

40-14x+x^2=2(x-4)√x.

Answer 1

$ODZ:x\geq0\\\\40-14x+x^{2} =2(x-4)\sqrt{x} \\\\x^{2} -14x+40=2\sqrt{x}(x-4)\\\\(x-4)(x-10)-2\sqrt{x}(x-4)=0\\\\(x-4)(x-10-2\sqrt{x})=0\\\\\left[\begin{array}{ccc}x-4=0\\x-10-2\sqrt{x} =0\end{array}\right\\\\1)x-4=0\Rightarrow x_{1}=4\\\\2)x-10-2\sqrt{x}=0\\\\\sqrt{x}=m,m\geq0\\\\m^{2} -2m-10=0\\\\D=(-2)^{2}-4*(-10)=4+40=44=(2\sqrt{11})^{2}\\\\m_{1}=\frac{2+2\sqrt{11}}{2}=1+\sqrt{11}\\\\m_{2}=\frac{2-2\sqrt{11}}{2}=1-\sqrt{11}$

$\sqrt{x}=1+\sqrt{11}\\\\x_{2} =(1+\sqrt{11})^{2}\\\\x_{2} =1+2\sqrt{11}+11=12+2\sqrt{11}\\\\Otvet:\boxed{4;12+2\sqrt{11}}$

Answer 2

To solve the given equation:

40 - 14x + x^2 = 2(x - 4)√x

Let's simplify the equation step by step:

1. Distribute the right side of the equation using the distributive property:

40 - 14x + x^2 = 2x√x - 8√x

2. Move all terms to one side of the equation to obtain a quadratic equation:

x^2 - (14x + 2x√x) + (8√x - 40) = 0

3. Rearrange the terms to simplify the equation further:

x^2 - 14x - 2x√x + 8√x - 40 = 0

4. Combine like terms:

x^2 - (14x - 2x√x + 8√x) - 40 = 0

x^2 - (14x + 2x√x + 8√x) - 40 = 0

x^2 - 14x - 10x√x - 40 = 0

5. The equation is quadratic in nature and includes a square root term. It is difficult to solve such equations algebraically. However, we can approximate the solutions using numerical methods or solve it graphically.

Numerical methods involve using iterative techniques to find approximate solutions. Graphically, you can plot the equation and find the points where the curve intersects the x-axis, which correspond to the solutions.

Please let me know if you would like an approximate solution using numerical methods or any further assistance.