Solve x+7/6+20/x-1=4

$$\frac{ x+7 }{ 6 } + \frac{ 20 }{ x-1 } =4$$

Solve for x (complex solution)

$$x=9+2\sqrt{14}i\approx 9+7.483314774i$$

$$x=-2\sqrt{14}i+9\approx 9-7.483314774i$$

Steps Using the Quadratic Formula

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

$$\left(x-1\right)\left(x+7\right)+6\times 20=24\left(x-1\right)$$

Use the distributive property to multiply x-1 by x+7 and combine like terms.

$$x^{2}+6x-7+6\times 20=24\left(x-1\right)$$

Multiply 6 and 20 to get 120.

$$x^{2}+6x-7+120=24\left(x-1\right)$$

Add -7 and 120 to get 113.

$$x^{2}+6x+113=24\left(x-1\right)$$

Use the distributive property to multiply 24 by x-1.

$$x^{2}+6x+113=24x-24$$

Subtract 24x from both sides.

$$x^{2}+6x+113-24x=-24$$

Combine 6x and -24x to get -18x.

$$x^{2}-18x+113=-24$$

Add 24 to both sides.

$$x^{2}-18x+113+24=0$$

Add 113 and 24 to get 137.

$$x^{2}-18x+137=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 137 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 137}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 137}}{2}$$

Multiply -4 times 137.

$$x=\frac{-\left(-18\right)±\sqrt{324-548}}{2}$$

Add 324 to -548.

$$x=\frac{-\left(-18\right)±\sqrt{-224}}{2}$$

Take the square root of -224.

$$x=\frac{-\left(-18\right)±4\sqrt{14}i}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±4\sqrt{14}i}{2}$$

Now solve the equation x=\frac{18±4\sqrt{14}i}{2} when ± is plus. Add 18 to 4i\sqrt{14}.

$$x=\frac{18+4\sqrt{14}i}{2}$$

Divide 18+4i\sqrt{14} by 2.

$$x=9+2\sqrt{14}i$$

Now solve the equation x=\frac{18±4\sqrt{14}i}{2} when ± is minus. Subtract 4i\sqrt{14} from 18.

$$x=\frac{-4\sqrt{14}i+18}{2}$$

Divide 18-4i\sqrt{14} by 2.

$$x=-2\sqrt{14}i+9$$

The equation is now solved.

$$x=9+2\sqrt{14}i x=-2\sqrt{14}i+9$$

Steps for Completing the Square

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

$$\left(x-1\right)\left(x+7\right)+6\times 20=24\left(x-1\right)$$

Use the distributive property to multiply x-1 by x+7 and combine like terms.

$$x^{2}+6x-7+6\times 20=24\left(x-1\right)$$

Multiply 6 and 20 to get 120.

$$x^{2}+6x-7+120=24\left(x-1\right)$$

Add -7 and 120 to get 113.

$$x^{2}+6x+113=24\left(x-1\right)$$

Use the distributive property to multiply 24 by x-1.

$$x^{2}+6x+113=24x-24$$

Subtract 24x from both sides.

$$x^{2}+6x+113-24x=-24$$

Combine 6x and -24x to get -18x.

$$x^{2}-18x+113=-24$$

Subtract 113 from both sides.

$$x^{2}-18x=-24-113$$

Subtract 113 from -24 to get -137.

$$x^{2}-18x=-137$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-137+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-137+81$$

Add -137 to 81.

$$x^{2}-18x+81=-56$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=-56$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-56}$$

Simplify.

$$x-9=2\sqrt{14}i x-9=-2\sqrt{14}i$$

Add 9 to both sides of the equation.

$$x=9+2\sqrt{14}i x=-2\sqrt{14}i+9$$