Solve 2/x-1-5/sqrt{x-1}+3=0

$$\frac { 2 } { x - 1 } - \frac { 5 } { \sqrt { x - 1 } } + 3 = 0$$

Solve for x

$$x = \frac{13}{9} = 1\frac{4}{9} \approx 1.444444444$$

$$x=2$$

Solution Steps

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$2-\left(x-1\right)^{\frac{1}{2}}\times 5+\left(x-1\right)\times 3=0$$

Use the distributive property to multiply x-1 by 3.

$$2-\left(x-1\right)^{\frac{1}{2}}\times 5+3x-3=0$$

Subtract 3 from 2 to get -1.

$$-1-\left(x-1\right)^{\frac{1}{2}}\times 5+3x=0$$

Reorder the terms.

$$3x-5\sqrt{x-1}-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$3x-5\sqrt{x-1}=1$$

Subtract 3x from both sides of the equation.

$$-5\sqrt{x-1}=1-3x$$

Square both sides of the equation.

$$\left(-5\sqrt{x-1}\right)^{2}=\left(1-3x\right)^{2}$$

Expand \left(-5\sqrt{x-1}\right)^{2}.

$$\left(-5\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(1-3x\right)^{2}$$

Calculate -5 to the power of 2 and get 25.

$$25\left(\sqrt{x-1}\right)^{2}=\left(1-3x\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$25\left(x-1\right)=\left(1-3x\right)^{2}$$

Use the distributive property to multiply 25 by x-1.

$$25x-25=\left(1-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-3x\right)^{2}.

$$25x-25=1-6x+9x^{2}$$

Add 6x to both sides.

$$25x-25+6x=1+9x^{2}$$

Combine 25x and 6x to get 31x.

$$31x-25=1+9x^{2}$$

Subtract 9x^{2} from both sides.

$$31x-25-9x^{2}=1$$

Subtract 1 from both sides.

$$31x-25-9x^{2}-1=0$$

Subtract 1 from -25 to get -26.

$$31x-26-9x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-9x^{2}+31x-26=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-26. To find a and b, set up a system to be solved.

$$a+b=31 ab=-9\left(-26\right)=234$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 234.

$$1,234 2,117 3,78 6,39 9,26 13,18$$

Calculate the sum for each pair.

$$1+234=235 2+117=119 3+78=81 6+39=45 9+26=35 13+18=31$$

The solution is the pair that gives sum 31.

$$a=18 b=13$$

Rewrite -9x^{2}+31x-26 as \left(-9x^{2}+18x\right)+\left(13x-26\right).

$$\left(-9x^{2}+18x\right)+\left(13x-26\right)$$

Factor out 9x in the first and -13 in the second group.

$$9x\left(-x+2\right)-13\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(9x-13\right)$$

To find equation solutions, solve -x+2=0 and 9x-13=0.

$$x=2 x=\frac{13}{9}$$

Substitute 2 for x in the equation \frac{2}{x-1}-\frac{5}{\sqrt{x-1}}+3=0.

$$\frac{2}{2-1}-\frac{5}{\sqrt{2-1}}+3=0$$

Simplify. The value x=2 satisfies the equation.

$$0=0$$

Substitute \frac{13}{9} for x in the equation \frac{2}{x-1}-\frac{5}{\sqrt{x-1}}+3=0.

$$\frac{2}{\frac{13}{9}-1}-\frac{5}{\sqrt{\frac{13}{9}-1}}+3=0$$

Simplify. The value x=\frac{13}{9} satisfies the equation.

$$0=0$$

List all solutions of -5\sqrt{x-1}=1-3x.

$$x=2 x=\frac{13}{9}$$