Solve 4(x-2)/5x-26=x+2/x-4

$$\frac { 4 ( x - 2 ) } { 5 x - 26 } = \frac { x + 2 } { x - 4 }$$

Solve for x

$$x=-14$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 4,\frac{26}{5} since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(5x-26\right), the least common multiple of 5x-26,x-4.

$$\left(x-4\right)\times 4\left(x-2\right)=\left(5x-26\right)\left(x+2\right)$$

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

$$\left(4x-16\right)\left(x-2\right)=\left(5x-26\right)\left(x+2\right)$$

Use the distributive property to multiply 4x-16 by x-2 and combine like terms.

$$4x^{2}-24x+32=\left(5x-26\right)\left(x+2\right)$$

Use the distributive property to multiply 5x-26 by x+2 and combine like terms.

$$4x^{2}-24x+32=5x^{2}-16x-52$$

Subtract 5x^{2} from both sides.

$$4x^{2}-24x+32-5x^{2}=-16x-52$$

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

$$-x^{2}-24x+32=-16x-52$$

Add 16x to both sides.

$$-x^{2}-24x+32+16x=-52$$

Combine -24x and 16x to get -8x.

$$-x^{2}-8x+32=-52$$

Add 52 to both sides.

$$-x^{2}-8x+32+52=0$$

Add 32 and 52 to get 84.

$$-x^{2}-8x+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 84}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 84}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\times 84}}{2\left(-1\right)}$$

Multiply 4 times 84.

$$x=\frac{-\left(-8\right)±\sqrt{64+336}}{2\left(-1\right)}$$

Add 64 to 336.

$$x=\frac{-\left(-8\right)±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$x=\frac{-\left(-8\right)±20}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±20}{-2}$$

Now solve the equation x=\frac{8±20}{-2} when ± is plus. Add 8 to 20.

$$x=\frac{28}{-2}$$

Divide 28 by -2.

$$x=-14$$

Now solve the equation x=\frac{8±20}{-2} when ± is minus. Subtract 20 from 8.

$$x=-\frac{12}{-2}$$

Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-14 x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 4,\frac{26}{5} since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(5x-26\right), the least common multiple of 5x-26,x-4.

$$\left(x-4\right)\times 4\left(x-2\right)=\left(5x-26\right)\left(x+2\right)$$

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

$$\left(4x-16\right)\left(x-2\right)=\left(5x-26\right)\left(x+2\right)$$

Use the distributive property to multiply 4x-16 by x-2 and combine like terms.

$$4x^{2}-24x+32=\left(5x-26\right)\left(x+2\right)$$

Use the distributive property to multiply 5x-26 by x+2 and combine like terms.

$$4x^{2}-24x+32=5x^{2}-16x-52$$

Subtract 5x^{2} from both sides.

$$4x^{2}-24x+32-5x^{2}=-16x-52$$

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

$$-x^{2}-24x+32=-16x-52$$

Add 16x to both sides.

$$-x^{2}-24x+32+16x=-52$$

Combine -24x and 16x to get -8x.

$$-x^{2}-8x+32=-52$$

Subtract 32 from both sides.

$$-x^{2}-8x=-52-32$$

Subtract 32 from -52 to get -84.

$$-x^{2}-8x=-84$$

Divide both sides by -1.

$$\frac{-x^{2}-8x}{-1}=-\frac{84}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=-\frac{84}{-1}$$

Divide -8 by -1.

$$x^{2}+8x=-\frac{84}{-1}$$

Divide -84 by -1.

$$x^{2}+8x=84$$

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

$$x^{2}+8x+4^{2}=84+4^{2}$$

Square 4.

$$x^{2}+8x+16=84+16$$

Add 84 to 16.

$$x^{2}+8x+16=100$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{100}$$

Simplify.

$$x+4=10 x+4=-10$$

Subtract 4 from both sides of the equation.

$$x=6 x=-14$$