Solve x+1/4-20/x-5=1

$$\frac { x + 1 } { 4 } - \frac { 20 } { x - 5 } = 1$$

Solve for x

$$x=-5$$

$$x=13$$

Steps Using the Quadratic Formula

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

$$\left(x-5\right)\left(x+1\right)-4\times 20=4\left(x-5\right)$$

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

$$x^{2}-4x-5-4\times 20=4\left(x-5\right)$$

Multiply -4 and 20 to get -80.

$$x^{2}-4x-5-80=4\left(x-5\right)$$

Subtract 80 from -5 to get -85.

$$x^{2}-4x-85=4\left(x-5\right)$$

Use the distributive property to multiply 4 by x-5.

$$x^{2}-4x-85=4x-20$$

Subtract 4x from both sides.

$$x^{2}-4x-85-4x=-20$$

Combine -4x and -4x to get -8x.

$$x^{2}-8x-85=-20$$

Add 20 to both sides.

$$x^{2}-8x-85+20=0$$

Add -85 and 20 to get -65.

$$x^{2}-8x-65=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -65 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-65\right)}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-65\right)}}{2}$$

Multiply -4 times -65.

$$x=\frac{-\left(-8\right)±\sqrt{64+260}}{2}$$

Add 64 to 260.

$$x=\frac{-\left(-8\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-8\right)±18}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±18}{2}$$

Now solve the equation x=\frac{8±18}{2} when ± is plus. Add 8 to 18.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{8±18}{2} when ± is minus. Subtract 18 from 8.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

The equation is now solved.

$$x=13 x=-5$$

Steps for Completing the Square

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

$$\left(x-5\right)\left(x+1\right)-4\times 20=4\left(x-5\right)$$

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

$$x^{2}-4x-5-4\times 20=4\left(x-5\right)$$

Multiply -4 and 20 to get -80.

$$x^{2}-4x-5-80=4\left(x-5\right)$$

Subtract 80 from -5 to get -85.

$$x^{2}-4x-85=4\left(x-5\right)$$

Use the distributive property to multiply 4 by x-5.

$$x^{2}-4x-85=4x-20$$

Subtract 4x from both sides.

$$x^{2}-4x-85-4x=-20$$

Combine -4x and -4x to get -8x.

$$x^{2}-8x-85=-20$$

Add 85 to both sides.

$$x^{2}-8x=-20+85$$

Add -20 and 85 to get 65.

$$x^{2}-8x=65$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=65+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=65+16$$

Add 65 to 16.

$$x^{2}-8x+16=81$$

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}.

$$\left(x-4\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-4=9 x-4=-9$$

Add 4 to both sides of the equation.

$$x=13 x=-5$$