Solve 2x/7x-4=1/x-1

$$\frac { 2 x } { 7 x - 4 } = \frac { 1 } { x - 1 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{4}{7},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(7x-4\right), the least common multiple of 7x-4,x-1.

$$\left(x-1\right)\times 2x=7x-4$$

Use the distributive property to multiply x-1 by 2.

$$\left(2x-2\right)x=7x-4$$

Use the distributive property to multiply 2x-2 by x.

$$2x^{2}-2x=7x-4$$

Subtract 7x from both sides.

$$2x^{2}-2x-7x=-4$$

Combine -2x and -7x to get -9x.

$$2x^{2}-9x=-4$$

Add 4 to both sides.

$$2x^{2}-9x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -9 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\times 4}}{2\times 2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\times 4}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\times 4}}{2\times 2}$$

Multiply -8 times 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-32}}{2\times 2}$$

Add 81 to -32.

$$x=\frac{-\left(-9\right)±\sqrt{49}}{2\times 2}$$

Take the square root of 49.

$$x=\frac{-\left(-9\right)±7}{2\times 2}$$

The opposite of -9 is 9.

$$x=\frac{9±7}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{9±7}{4}$$

Now solve the equation x=\frac{9±7}{4} when ± is plus. Add 9 to 7.

$$x=\frac{16}{4}$$

Divide 16 by 4.

$$x=4$$

Now solve the equation x=\frac{9±7}{4} when ± is minus. Subtract 7 from 9.

$$x=\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=4 x=\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{4}{7},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(7x-4\right), the least common multiple of 7x-4,x-1.

$$\left(x-1\right)\times 2x=7x-4$$

Use the distributive property to multiply x-1 by 2.

$$\left(2x-2\right)x=7x-4$$

Use the distributive property to multiply 2x-2 by x.

$$2x^{2}-2x=7x-4$$

Subtract 7x from both sides.

$$2x^{2}-2x-7x=-4$$

Combine -2x and -7x to get -9x.

$$2x^{2}-9x=-4$$

Divide both sides by 2.

$$\frac{2x^{2}-9x}{2}=-\frac{4}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{9}{2}x=-\frac{4}{2}$$

Divide -4 by 2.

$$x^{2}-\frac{9}{2}x=-2$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-2+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-2+\frac{81}{16}$$

Add -2 to \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{49}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{7}{4} x-\frac{9}{4}=-\frac{7}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=4 x=\frac{1}{2}$$