Solve 1/2x-1+2/5=2/x-6

$$\frac { 1 } { 2 x - 1 } + \frac { 2 } { 5 } = \frac { 2 } { x - 6 }$$

Solve for x

$$x = \frac{3 \sqrt{201} + 41}{8} \approx 10.44154258$$

$$x=\frac{41-3\sqrt{201}}{8}\approx -0.19154258$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{2},6 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-6\right)\left(2x-1\right), the least common multiple of 2x-1,5,x-6.

$$5x-30+5\left(x-6\right)\left(2x-1\right)\times \frac{2}{5}=\left(10x-5\right)\times 2$$

Multiply 5 and \frac{2}{5} to get 2.

$$5x-30+2\left(x-6\right)\left(2x-1\right)=\left(10x-5\right)\times 2$$

Use the distributive property to multiply 2 by x-6.

$$5x-30+\left(2x-12\right)\left(2x-1\right)=\left(10x-5\right)\times 2$$

Use the distributive property to multiply 2x-12 by 2x-1 and combine like terms.

$$5x-30+4x^{2}-26x+12=\left(10x-5\right)\times 2$$

Combine 5x and -26x to get -21x.

$$-21x-30+4x^{2}+12=\left(10x-5\right)\times 2$$

Add -30 and 12 to get -18.

$$-21x-18+4x^{2}=\left(10x-5\right)\times 2$$

Use the distributive property to multiply 10x-5 by 2.

$$-21x-18+4x^{2}=20x-10$$

Subtract 20x from both sides.

$$-21x-18+4x^{2}-20x=-10$$

Combine -21x and -20x to get -41x.

$$-41x-18+4x^{2}=-10$$

Add 10 to both sides.

$$-41x-18+4x^{2}+10=0$$

Add -18 and 10 to get -8.

$$-41x-8+4x^{2}=0$$

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.

$$4x^{2}-41x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -41 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 4\left(-8\right)}}{2\times 4}$$

Square -41.

$$x=\frac{-\left(-41\right)±\sqrt{1681-4\times 4\left(-8\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-41\right)±\sqrt{1681-16\left(-8\right)}}{2\times 4}$$

Multiply -16 times -8.

$$x=\frac{-\left(-41\right)±\sqrt{1681+128}}{2\times 4}$$

Add 1681 to 128.

$$x=\frac{-\left(-41\right)±\sqrt{1809}}{2\times 4}$$

Take the square root of 1809.

$$x=\frac{-\left(-41\right)±3\sqrt{201}}{2\times 4}$$

The opposite of -41 is 41.

$$x=\frac{41±3\sqrt{201}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{41±3\sqrt{201}}{8}$$

Now solve the equation x=\frac{41±3\sqrt{201}}{8} when ± is plus. Add 41 to 3\sqrt{201}.

$$x=\frac{3\sqrt{201}+41}{8}$$

Now solve the equation x=\frac{41±3\sqrt{201}}{8} when ± is minus. Subtract 3\sqrt{201} from 41.

$$x=\frac{41-3\sqrt{201}}{8}$$

The equation is now solved.

$$x=\frac{3\sqrt{201}+41}{8} x=\frac{41-3\sqrt{201}}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{2},6 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-6\right)\left(2x-1\right), the least common multiple of 2x-1,5,x-6.

$$5x-30+5\left(x-6\right)\left(2x-1\right)\times \frac{2}{5}=\left(10x-5\right)\times 2$$

Multiply 5 and \frac{2}{5} to get 2.

$$5x-30+2\left(x-6\right)\left(2x-1\right)=\left(10x-5\right)\times 2$$

Use the distributive property to multiply 2 by x-6.

$$5x-30+\left(2x-12\right)\left(2x-1\right)=\left(10x-5\right)\times 2$$

Use the distributive property to multiply 2x-12 by 2x-1 and combine like terms.

$$5x-30+4x^{2}-26x+12=\left(10x-5\right)\times 2$$

Combine 5x and -26x to get -21x.

$$-21x-30+4x^{2}+12=\left(10x-5\right)\times 2$$

Add -30 and 12 to get -18.

$$-21x-18+4x^{2}=\left(10x-5\right)\times 2$$

Use the distributive property to multiply 10x-5 by 2.

$$-21x-18+4x^{2}=20x-10$$

Subtract 20x from both sides.

$$-21x-18+4x^{2}-20x=-10$$

Combine -21x and -20x to get -41x.

$$-41x-18+4x^{2}=-10$$

Add 18 to both sides.

$$-41x+4x^{2}=-10+18$$

Add -10 and 18 to get 8.

$$-41x+4x^{2}=8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}-41x=8$$

Divide both sides by 4.

$$\frac{4x^{2}-41x}{4}=\frac{8}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{41}{4}x=\frac{8}{4}$$

Divide 8 by 4.

$$x^{2}-\frac{41}{4}x=2$$

Divide -\frac{41}{4}, the coefficient of the x term, by 2 to get -\frac{41}{8}. Then add the square of -\frac{41}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{41}{4}x+\left(-\frac{41}{8}\right)^{2}=2+\left(-\frac{41}{8}\right)^{2}$$

Square -\frac{41}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{41}{4}x+\frac{1681}{64}=2+\frac{1681}{64}$$

Add 2 to \frac{1681}{64}.

$$x^{2}-\frac{41}{4}x+\frac{1681}{64}=\frac{1809}{64}$$

Factor x^{2}-\frac{41}{4}x+\frac{1681}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{41}{8}\right)^{2}=\frac{1809}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{41}{8}\right)^{2}}=\sqrt{\frac{1809}{64}}$$

Simplify.

$$x-\frac{41}{8}=\frac{3\sqrt{201}}{8} x-\frac{41}{8}=-\frac{3\sqrt{201}}{8}$$

Add \frac{41}{8} to both sides of the equation.

$$x=\frac{3\sqrt{201}+41}{8} x=\frac{41-3\sqrt{201}}{8}$$