√7x-5-√3x-2=√4x-3 equation

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  _____         _____         _____    
\/ 7*x  - 5 - \/ 3*x  - 2 = \/ 4*x  - 3

\left(- \sqrt{3 x} + \left(\sqrt{7 x} - 5\right)\right) - 2 = \sqrt{4 x} - 3

Simplify

Detail solution

Given the equation

\left(- \sqrt{3 x} + \left(\sqrt{7 x} - 5\right)\right) - 2 = \sqrt{4 x} - 3

Transfer the right side of the equation left part with negative sign

\sqrt{x} \left(-2 - \sqrt{3} + \sqrt{7}\right) = 4

We raise the equation sides to 2-th degree

x \left(-2 - \sqrt{3} + \sqrt{7}\right)^{2} = 16

x \left(-2 - \sqrt{3} + \sqrt{7}\right)^{2} = 16

Transfer the right side of the equation left part with negative sign

x \left(-2 - \sqrt{3} + \sqrt{7}\right)^{2} - 16 = 0

Expand brackets in the left part

-16 + x-2+sqrt+7 - sqrt3)^2 = 0

Looking for similar summands in the left part:

-16 + x*(-2 + sqrt(7) - sqrt(3))^2 = 0

Move free summands (without x)
from left part to right part, we given:

x \left(-2 - \sqrt{3} + \sqrt{7}\right)^{2} = 16

Divide both parts of the equation by (-2 + sqrt(7) - sqrt(3))^2

x = 16 / ((-2 + sqrt(7) - sqrt(3))^2)

We get the answer: x = 16/(2 + sqrt(3) - sqrt(7))^2

Because

\sqrt{x} = \frac{4}{-2 - \sqrt{3} + \sqrt{7}}

and

\sqrt{x} \geq 0

then

\frac{4}{-2 - \sqrt{3} + \sqrt{7}} \geq 0

The final answer:
This equation has no roots

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Sum and product of roots [src]

sum

0

0

product

1

1