Solve 3x/x+4-3=x-18/x-4

$$\frac { 3 x } { x + 4 } - 3 = \frac { x - 18 } { x - 4 }$$

Solve for x

$$x=-10$$

$$x=12$$

Steps Using the Quadratic Formula

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

$$\left(x-4\right)\times 3x+\left(x-4\right)\left(x+4\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply x-4 by 3.

$$\left(3x-12\right)x+\left(x-4\right)\left(x+4\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply 3x-12 by x.

$$3x^{2}-12x+\left(x-4\right)\left(x+4\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

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

$$3x^{2}-12x+\left(x^{2}-16\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply x^{2}-16 by -3.

$$3x^{2}-12x-3x^{2}+48=\left(x+4\right)\left(x-18\right)$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-12x+48=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply x+4 by x-18 and combine like terms.

$$-12x+48=x^{2}-14x-72$$

Subtract x^{2} from both sides.

$$-12x+48-x^{2}=-14x-72$$

Add 14x to both sides.

$$-12x+48-x^{2}+14x=-72$$

Combine -12x and 14x to get 2x.

$$2x+48-x^{2}=-72$$

Add 72 to both sides.

$$2x+48-x^{2}+72=0$$

Add 48 and 72 to get 120.

$$2x+120-x^{2}=0$$

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.

$$-x^{2}+2x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 120}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 120}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\times 120}}{2\left(-1\right)}$$

Multiply 4 times 120.

$$x=\frac{-2±\sqrt{4+480}}{2\left(-1\right)}$$

Add 4 to 480.

$$x=\frac{-2±\sqrt{484}}{2\left(-1\right)}$$

Take the square root of 484.

$$x=\frac{-2±22}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±22}{-2}$$

Now solve the equation x=\frac{-2±22}{-2} when ± is plus. Add -2 to 22.

$$x=\frac{20}{-2}$$

Divide 20 by -2.

$$x=-10$$

Now solve the equation x=\frac{-2±22}{-2} when ± is minus. Subtract 22 from -2.

$$x=-\frac{24}{-2}$$

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=-10 x=12$$

Steps for Completing the Square

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

$$\left(x-4\right)\times 3x+\left(x-4\right)\left(x+4\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply x-4 by 3.

$$\left(3x-12\right)x+\left(x-4\right)\left(x+4\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply 3x-12 by x.

$$3x^{2}-12x+\left(x-4\right)\left(x+4\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

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

$$3x^{2}-12x+\left(x^{2}-16\right)\left(-3\right)=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply x^{2}-16 by -3.

$$3x^{2}-12x-3x^{2}+48=\left(x+4\right)\left(x-18\right)$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-12x+48=\left(x+4\right)\left(x-18\right)$$

Use the distributive property to multiply x+4 by x-18 and combine like terms.

$$-12x+48=x^{2}-14x-72$$

Subtract x^{2} from both sides.

$$-12x+48-x^{2}=-14x-72$$

Add 14x to both sides.

$$-12x+48-x^{2}+14x=-72$$

Combine -12x and 14x to get 2x.

$$2x+48-x^{2}=-72$$

Subtract 48 from both sides.

$$2x-x^{2}=-72-48$$

Subtract 48 from -72 to get -120.

$$2x-x^{2}=-120$$

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.

$$-x^{2}+2x=-120$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{120}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=-\frac{120}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=-\frac{120}{-1}$$

Divide -120 by -1.

$$x^{2}-2x=120$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=120+1$$

Add 120 to 1.

$$x^{2}-2x+1=121$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=121$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{121}$$

Simplify.

$$x-1=11 x-1=-11$$

Add 1 to both sides of the equation.

$$x=12 x=-10$$