Solve 7/(x-1)-2/sqrt{x-1}+1/7=0

$$\frac{ 7 }{ (x-1) } - \frac{ 2 }{ \sqrt{ x-1 } } + \frac{ 1 }{ 7 } =0$$

Solve for x

$$x=50$$

Solution Steps

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 7\left(x-1\right), the least common multiple of x-1,7.

$$7\times 7-7\left(x-1\right)^{\frac{1}{2}}\times 2+7\left(x-1\right)\times \frac{1}{7}=0$$

Multiply 7 and 7 to get 49.

$$49-7\left(x-1\right)^{\frac{1}{2}}\times 2+7\left(x-1\right)\times \frac{1}{7}=0$$

Multiply 7 and 2 to get 14.

$$49-14\left(x-1\right)^{\frac{1}{2}}+7\left(x-1\right)\times \frac{1}{7}=0$$

Multiply 7 and \frac{1}{7} to get 1.

$$49-14\left(x-1\right)^{\frac{1}{2}}+x-1=0$$

Subtract 1 from 49 to get 48.

$$48-14\left(x-1\right)^{\frac{1}{2}}+x=0$$

Reorder the terms.

$$x-14\sqrt{x-1}+48=0$$

Subtract 48 from both sides. Anything subtracted from zero gives its negation.

$$x-14\sqrt{x-1}=-48$$

Subtract x from both sides of the equation.

$$-14\sqrt{x-1}=-48-x$$

Square both sides of the equation.

$$\left(-14\sqrt{x-1}\right)^{2}=\left(-48-x\right)^{2}$$

Expand \left(-14\sqrt{x-1}\right)^{2}.

$$\left(-14\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(-48-x\right)^{2}$$

Calculate -14 to the power of 2 and get 196.

$$196\left(\sqrt{x-1}\right)^{2}=\left(-48-x\right)^{2}$$

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

$$196\left(x-1\right)=\left(-48-x\right)^{2}$$

Use the distributive property to multiply 196 by x-1.

$$196x-196=\left(-48-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-48-x\right)^{2}.

$$196x-196=2304+96x+x^{2}$$

Subtract 96x from both sides.

$$196x-196-96x=2304+x^{2}$$

Combine 196x and -96x to get 100x.

$$100x-196=2304+x^{2}$$

Subtract x^{2} from both sides.

$$100x-196-x^{2}=2304$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+100x-196=2304$$

Subtract 2304 from both sides of the equation.

$$-x^{2}+100x-196-2304=2304-2304$$

Subtracting 2304 from itself leaves 0.

$$-x^{2}+100x-196-2304=0$$

Subtract 2304 from -196.

$$-x^{2}+100x-2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 100 for b, and -2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-1\right)\left(-2500\right)}}{2\left(-1\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-1\right)\left(-2500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-100±\sqrt{10000+4\left(-2500\right)}}{2\left(-1\right)}$$

Multiply 4 times -2500.

$$x=\frac{-100±\sqrt{10000-10000}}{2\left(-1\right)}$$

Add 10000 to -10000.

$$x=\frac{-100±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{100}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{100}{-2}$$

Divide -100 by -2.

$$x=50$$

Substitute 50 for x in the equation \frac{7}{x-1}-\frac{2}{\sqrt{x-1}}+\frac{1}{7}=0.

$$\frac{7}{50-1}-\frac{2}{\sqrt{50-1}}+\frac{1}{7}=0$$

Simplify. The value x=50 satisfies the equation.

$$0=0$$

Equation -14\sqrt{x-1}=-x-48 has a unique solution.

$$x=50$$