Solve x/x-2+4/x-1=5

$$\frac { x } { x - 2 } + \frac { 4 } { x - 1 } = 5$$

Solve for x

$$x=3$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right), the least common multiple of x-2,x-1.

$$\left(x-1\right)x+\left(x-2\right)\times 4=5\left(x-2\right)\left(x-1\right)$$

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

$$x^{2}-x+\left(x-2\right)\times 4=5\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-2 by 4.

$$x^{2}-x+4x-8=5\left(x-2\right)\left(x-1\right)$$

Combine -x and 4x to get 3x.

$$x^{2}+3x-8=5\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 5 by x-2.

$$x^{2}+3x-8=\left(5x-10\right)\left(x-1\right)$$

Use the distributive property to multiply 5x-10 by x-1 and combine like terms.

$$x^{2}+3x-8=5x^{2}-15x+10$$

Subtract 5x^{2} from both sides.

$$x^{2}+3x-8-5x^{2}=-15x+10$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+3x-8=-15x+10$$

Add 15x to both sides.

$$-4x^{2}+3x-8+15x=10$$

Combine 3x and 15x to get 18x.

$$-4x^{2}+18x-8=10$$

Subtract 10 from both sides.

$$-4x^{2}+18x-8-10=0$$

Subtract 10 from -8 to get -18.

$$-4x^{2}+18x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 18 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-4\right)\left(-18\right)}}{2\left(-4\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-4\right)\left(-18\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-18±\sqrt{324+16\left(-18\right)}}{2\left(-4\right)}$$

Multiply 16 times -18.

$$x=\frac{-18±\sqrt{324-288}}{2\left(-4\right)}$$

Add 324 to -288.

$$x=\frac{-18±\sqrt{36}}{2\left(-4\right)}$$

Take the square root of 36.

$$x=\frac{-18±6}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-18±6}{-8}$$

Now solve the equation x=\frac{-18±6}{-8} when ± is plus. Add -18 to 6.

$$x=-\frac{12}{-8}$$

Reduce the fraction \frac{-12}{-8} to lowest terms by extracting and canceling out 4.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-18±6}{-8} when ± is minus. Subtract 6 from -18.

$$x=-\frac{24}{-8}$$

Divide -24 by -8.

$$x=3$$

The equation is now solved.

$$x=\frac{3}{2} x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right), the least common multiple of x-2,x-1.

$$\left(x-1\right)x+\left(x-2\right)\times 4=5\left(x-2\right)\left(x-1\right)$$

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

$$x^{2}-x+\left(x-2\right)\times 4=5\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-2 by 4.

$$x^{2}-x+4x-8=5\left(x-2\right)\left(x-1\right)$$

Combine -x and 4x to get 3x.

$$x^{2}+3x-8=5\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 5 by x-2.

$$x^{2}+3x-8=\left(5x-10\right)\left(x-1\right)$$

Use the distributive property to multiply 5x-10 by x-1 and combine like terms.

$$x^{2}+3x-8=5x^{2}-15x+10$$

Subtract 5x^{2} from both sides.

$$x^{2}+3x-8-5x^{2}=-15x+10$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+3x-8=-15x+10$$

Add 15x to both sides.

$$-4x^{2}+3x-8+15x=10$$

Combine 3x and 15x to get 18x.

$$-4x^{2}+18x-8=10$$

Add 8 to both sides.

$$-4x^{2}+18x=10+8$$

Add 10 and 8 to get 18.

$$-4x^{2}+18x=18$$

Divide both sides by -4.

$$\frac{-4x^{2}+18x}{-4}=\frac{18}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{18}{-4}x=\frac{18}{-4}$$

Reduce the fraction \frac{18}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{9}{2}x=\frac{18}{-4}$$

Reduce the fraction \frac{18}{-4} to lowest terms by extracting and canceling out 2.

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

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.

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

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

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

Add -\frac{9}{2} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{9}{4}=\frac{3}{4} x-\frac{9}{4}=-\frac{3}{4}$$

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

$$x=3 x=\frac{3}{2}$$