Solve 4x/x+3-4=x-23/x-3

$$\frac { 4 x } { x + 3 } - 4 = \frac { x - 23 } { x - 3 }$$

Solve for x

$$x=-7$$

$$x=15$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3.

$$\left(x-3\right)\times 4x+\left(x-3\right)\left(x+3\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

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

$$\left(4x-12\right)x+\left(x-3\right)\left(x+3\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply 4x-12 by x.

$$4x^{2}-12x+\left(x-3\right)\left(x+3\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply x-3 by x+3 and combine like terms.

$$4x^{2}-12x+\left(x^{2}-9\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply x^{2}-9 by -4.

$$4x^{2}-12x-4x^{2}+36=\left(x+3\right)\left(x-23\right)$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-12x+36=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply x+3 by x-23 and combine like terms.

$$-12x+36=x^{2}-20x-69$$

Subtract x^{2} from both sides.

$$-12x+36-x^{2}=-20x-69$$

Add 20x to both sides.

$$-12x+36-x^{2}+20x=-69$$

Combine -12x and 20x to get 8x.

$$8x+36-x^{2}=-69$$

Add 69 to both sides.

$$8x+36-x^{2}+69=0$$

Add 36 and 69 to get 105.

$$8x+105-x^{2}=0$$

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.

$$-x^{2}+8x+105=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and 105 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\times 105}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\times 105}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\times 105}}{2\left(-1\right)}$$

Multiply 4 times 105.

$$x=\frac{-8±\sqrt{64+420}}{2\left(-1\right)}$$

Add 64 to 420.

$$x=\frac{-8±\sqrt{484}}{2\left(-1\right)}$$

Take the square root of 484.

$$x=\frac{-8±22}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±22}{-2}$$

Now solve the equation x=\frac{-8±22}{-2} when ± is plus. Add -8 to 22.

$$x=\frac{14}{-2}$$

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{-8±22}{-2} when ± is minus. Subtract 22 from -8.

$$x=-\frac{30}{-2}$$

Divide -30 by -2.

$$x=15$$

The equation is now solved.

$$x=-7 x=15$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x+3,x-3.

$$\left(x-3\right)\times 4x+\left(x-3\right)\left(x+3\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

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

$$\left(4x-12\right)x+\left(x-3\right)\left(x+3\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply 4x-12 by x.

$$4x^{2}-12x+\left(x-3\right)\left(x+3\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply x-3 by x+3 and combine like terms.

$$4x^{2}-12x+\left(x^{2}-9\right)\left(-4\right)=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply x^{2}-9 by -4.

$$4x^{2}-12x-4x^{2}+36=\left(x+3\right)\left(x-23\right)$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-12x+36=\left(x+3\right)\left(x-23\right)$$

Use the distributive property to multiply x+3 by x-23 and combine like terms.

$$-12x+36=x^{2}-20x-69$$

Subtract x^{2} from both sides.

$$-12x+36-x^{2}=-20x-69$$

Add 20x to both sides.

$$-12x+36-x^{2}+20x=-69$$

Combine -12x and 20x to get 8x.

$$8x+36-x^{2}=-69$$

Subtract 36 from both sides.

$$8x-x^{2}=-69-36$$

Subtract 36 from -69 to get -105.

$$8x-x^{2}=-105$$

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.

$$-x^{2}+8x=-105$$

Divide both sides by -1.

$$\frac{-x^{2}+8x}{-1}=-\frac{105}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{8}{-1}x=-\frac{105}{-1}$$

Divide 8 by -1.

$$x^{2}-8x=-\frac{105}{-1}$$

Divide -105 by -1.

$$x^{2}-8x=105$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=105+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=105+16$$

Add 105 to 16.

$$x^{2}-8x+16=121$$

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}.

$$\left(x-4\right)^{2}=121$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{121}$$

Simplify.

$$x-4=11 x-4=-11$$

Add 4 to both sides of the equation.

$$x=15 x=-7$$