What is the answer to (6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1 ?

The answer to (6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1 is x=23.56067…

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(6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1

\frac{6 x ^{2}}{( 2 x + 1 + 1 ) ^{2}} = 2 x + x - 1 + 1

x = 23.56067 \dots

Solution steps

\frac{6 x ^{2}}{( 2 x + 1 + 1 ) ^{2}} = 2 x + x - 1 + 1

Multiply both sides by

(2 x + 1 + 1)^{2}

\frac{6 x ^{2}}{( 2 x + 1 + 1 ) ^{2}} (2 x + 1 + 1)^{2} = 2 x (2 x + 1 + 1)^{2} + x - 1 (2 x + 1 + 1)^{2} + 1 \cdot (2 x + 1 + 1)^{2}

Simplify

6 x^{2} = 2 x (2 x + 1 + 1)^{2} + x - 1 (2 x + 1 + 1)^{2} + (2 x + 1 + 1)^{2}

Expand

2 x (2 x + 1 + 1)^{2} + x - 1 (2 x + 1 + 1)^{2} + (2 x + 1 + 1)^{2} : 4 x^{2} + 6 x + 4 x 2 x + 1 + 2 x x - 1 + 2 x - 1 + 2 x - 1 2 x + 1 + 2 2 x + 1 + 2

6 x^{2} = 4 x^{2} + 6 x + 4 x 2 x + 1 + 2 x x - 1 + 2 x - 1 + 2 x - 1 2 x + 1 + 2 2 x + 1 + 2

Rewrite the equation with

x - 1 = u

and

x = u^{2} + 1

6 (u^{2} + 1)^{2} = 4 (u^{2} + 1)^{2} + 6 (u^{2} + 1) + 4 (u^{2} + 1) 2 (u^{2} + 1) + 1 + 2 (u^{2} + 1) u + 2 u + 2 u 2 (u^{2} + 1) + 1 + 2 2 (u^{2} + 1) + 1 + 2

Solve

6 (u^{2} + 1)^{2} = 4 (u^{2} + 1)^{2} + 6 (u^{2} + 1) + 4 (u^{2} + 1) 2 (u^{2} + 1) + 1 + 2 (u^{2} + 1) u + 2 u + 2 u 2 (u^{2} + 1) + 1 + 2 2 (u^{2} + 1) + 1 + 2 : u \approx - 2.52756 \dots, u \approx 4.74980 \dots

u \approx - 2.52756 \dots, u \approx 4.74980 \dots

Substitute back

u = x - 1,

solve for

x

Solve

x - 1 = - 2.52756 \dots :

No Solution for

x \in R

Solve

x - 1 = 4.74980 \dots : x = 23.56067 \dots

x = 23.56067 \dots

Verify Solutions:

x = 23.56067 \dots

True

The solution is

x = 23.56067 \dots

c = \frac{1}{( x ^{2} + x - 2 )}

2 x^{2} + 2 5 x - 5 = 0

(\frac{x - 1}{x})^{\frac{1}{2}} + (\frac{1 - 1}{x})^{\frac{1}{2}} = x

x + 1 + 2 x + 3 = 5

2 x - 3 = - 1

What is the answer to (6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1 ?
The answer to (6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1 is x=23.56067…
Solve for x: (6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1
The solution is x=23.56067…