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

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

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 \frac{1}{2},\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-5\right)\left(2x-1\right), the least common multiple of 2x-5,2x-1.

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

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

$$2x-1+10x-25=2\left(2x-5\right)\left(2x-1\right)$$

Combine 2x and 10x to get 12x.

$$12x-1-25=2\left(2x-5\right)\left(2x-1\right)$$

Subtract 25 from -1 to get -26.

$$12x-26=2\left(2x-5\right)\left(2x-1\right)$$

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

$$12x-26=\left(4x-10\right)\left(2x-1\right)$$

Use the distributive property to multiply 4x-10 by 2x-1 and combine like terms.

$$12x-26=8x^{2}-24x+10$$

Subtract 8x^{2} from both sides.

$$12x-26-8x^{2}=-24x+10$$

Add 24x to both sides.

$$12x-26-8x^{2}+24x=10$$

Combine 12x and 24x to get 36x.

$$36x-26-8x^{2}=10$$

Subtract 10 from both sides.

$$36x-26-8x^{2}-10=0$$

Subtract 10 from -26 to get -36.

$$36x-36-8x^{2}=0$$

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.

$$-8x^{2}+36x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 36 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\left(-8\right)\left(-36\right)}}{2\left(-8\right)}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\left(-8\right)\left(-36\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-36±\sqrt{1296+32\left(-36\right)}}{2\left(-8\right)}$$

Multiply 32 times -36.

$$x=\frac{-36±\sqrt{1296-1152}}{2\left(-8\right)}$$

Add 1296 to -1152.

$$x=\frac{-36±\sqrt{144}}{2\left(-8\right)}$$

Take the square root of 144.

$$x=\frac{-36±12}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-36±12}{-16}$$

Now solve the equation x=\frac{-36±12}{-16} when ± is plus. Add -36 to 12.

$$x=-\frac{24}{-16}$$

Reduce the fraction \frac{-24}{-16} to lowest terms by extracting and canceling out 8.

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

Now solve the equation x=\frac{-36±12}{-16} when ± is minus. Subtract 12 from -36.

$$x=-\frac{48}{-16}$$

Divide -48 by -16.

$$x=3$$

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 \frac{1}{2},\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-5\right)\left(2x-1\right), the least common multiple of 2x-5,2x-1.

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

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

$$2x-1+10x-25=2\left(2x-5\right)\left(2x-1\right)$$

Combine 2x and 10x to get 12x.

$$12x-1-25=2\left(2x-5\right)\left(2x-1\right)$$

Subtract 25 from -1 to get -26.

$$12x-26=2\left(2x-5\right)\left(2x-1\right)$$

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

$$12x-26=\left(4x-10\right)\left(2x-1\right)$$

Use the distributive property to multiply 4x-10 by 2x-1 and combine like terms.

$$12x-26=8x^{2}-24x+10$$

Subtract 8x^{2} from both sides.

$$12x-26-8x^{2}=-24x+10$$

Add 24x to both sides.

$$12x-26-8x^{2}+24x=10$$

Combine 12x and 24x to get 36x.

$$36x-26-8x^{2}=10$$

Add 26 to both sides.

$$36x-8x^{2}=10+26$$

Add 10 and 26 to get 36.

$$36x-8x^{2}=36$$

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.

$$-8x^{2}+36x=36$$

Divide both sides by -8.

$$\frac{-8x^{2}+36x}{-8}=\frac{36}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{36}{-8}x=\frac{36}{-8}$$

Reduce the fraction \frac{36}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{9}{2}x=\frac{36}{-8}$$

Reduce the fraction \frac{36}{-8} to lowest terms by extracting and canceling out 4.

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

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.

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