Solve x+4/x+2=8/4-x

$$\frac { x + 4 } { x + 2 } = \frac { 8 } { 4 - x }$$

Solve for x

$$x=-8$$

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x+2,4-x.

$$\left(x-4\right)\left(x+4\right)=-\left(2+x\right)\times 8$$

Consider \left(x-4\right)\left(x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$x^{2}-16=-\left(2+x\right)\times 8$$

Multiply -1 and 8 to get -8.

$$x^{2}-16=-8\left(2+x\right)$$

Use the distributive property to multiply -8 by 2+x.

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

Subtract -16 from both sides.

$$x^{2}-16-\left(-16\right)=-8x$$

The opposite of -16 is 16.

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

Add 8x to both sides.

$$x^{2}-16+16+8x=0$$

Add -16 and 16 to get 0.

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

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

$$x=\frac{-8±\sqrt{8^{2}}}{2}$$

Take the square root of 8^{2}.

$$x=\frac{-8±8}{2}$$

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

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

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

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=0 x=-8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x+2,4-x.

$$\left(x-4\right)\left(x+4\right)=-\left(2+x\right)\times 8$$

Consider \left(x-4\right)\left(x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$x^{2}-16=-\left(2+x\right)\times 8$$

Multiply -1 and 8 to get -8.

$$x^{2}-16=-8\left(2+x\right)$$

Use the distributive property to multiply -8 by 2+x.

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

Add 8x to both sides.

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

Add 16 to both sides.

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

Add -16 and 16 to get 0.

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

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.

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

Square 4.

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

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}.

$$\left(x+4\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{16}$$

Simplify.

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

Subtract 4 from both sides of the equation.

$$x=0 x=-8$$