Solve 4-(2*x+1/x+4)*(x+4)=16

$$4 - ( 2 \times \frac { x + 1 } { x + 4 } ) \times ( x + 4 ) = 16$$

Solve for x

$$x=-7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$\left(x+4\right)\times 4-2\left(x+1\right)\left(x+4\right)=16\left(x+4\right)$$

Use the distributive property to multiply x+4 by 4.

$$4x+16-2\left(x+1\right)\left(x+4\right)=16\left(x+4\right)$$

Use the distributive property to multiply 16 by x+4.

$$4x+16-2\left(x+1\right)\left(x+4\right)=16x+64$$

Subtract 16x from both sides.

$$4x+16-2\left(x+1\right)\left(x+4\right)-16x=64$$

Subtract 64 from both sides.

$$4x+16-2\left(x+1\right)\left(x+4\right)-16x-64=0$$

Use the distributive property to multiply -2 by x+1.

$$4x+16+\left(-2x-2\right)\left(x+4\right)-16x-64=0$$

Use the distributive property to multiply -2x-2 by x+4 and combine like terms.

$$4x+16-2x^{2}-10x-8-16x-64=0$$

Combine 4x and -10x to get -6x.

$$-6x+16-2x^{2}-8-16x-64=0$$

Subtract 8 from 16 to get 8.

$$-6x+8-2x^{2}-16x-64=0$$

Combine -6x and -16x to get -22x.

$$-22x+8-2x^{2}-64=0$$

Subtract 64 from 8 to get -56.

$$-22x-56-2x^{2}=0$$

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.

$$-2x^{2}-22x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -22 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-2\right)\left(-56\right)}}{2\left(-2\right)}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\left(-2\right)\left(-56\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-22\right)±\sqrt{484+8\left(-56\right)}}{2\left(-2\right)}$$

Multiply 8 times -56.

$$x=\frac{-\left(-22\right)±\sqrt{484-448}}{2\left(-2\right)}$$

Add 484 to -448.

$$x=\frac{-\left(-22\right)±\sqrt{36}}{2\left(-2\right)}$$

Take the square root of 36.

$$x=\frac{-\left(-22\right)±6}{2\left(-2\right)}$$

The opposite of -22 is 22.

$$x=\frac{22±6}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{22±6}{-4}$$

Now solve the equation x=\frac{22±6}{-4} when ± is plus. Add 22 to 6.

$$x=\frac{28}{-4}$$

Divide 28 by -4.

$$x=-7$$

Now solve the equation x=\frac{22±6}{-4} when ± is minus. Subtract 6 from 22.

$$x=\frac{16}{-4}$$

Divide 16 by -4.

$$x=-4$$

The equation is now solved.

$$x=-7 x=-4$$

Variable x cannot be equal to -4.

$$x=-7$$

Steps for Completing the Square

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$\left(x+4\right)\times 4-2\left(x+1\right)\left(x+4\right)=16\left(x+4\right)$$

Use the distributive property to multiply x+4 by 4.

$$4x+16-2\left(x+1\right)\left(x+4\right)=16\left(x+4\right)$$

Use the distributive property to multiply 16 by x+4.

$$4x+16-2\left(x+1\right)\left(x+4\right)=16x+64$$

Subtract 16x from both sides.

$$4x+16-2\left(x+1\right)\left(x+4\right)-16x=64$$

Use the distributive property to multiply -2 by x+1.

$$4x+16+\left(-2x-2\right)\left(x+4\right)-16x=64$$

Use the distributive property to multiply -2x-2 by x+4 and combine like terms.

$$4x+16-2x^{2}-10x-8-16x=64$$

Combine 4x and -10x to get -6x.

$$-6x+16-2x^{2}-8-16x=64$$

Subtract 8 from 16 to get 8.

$$-6x+8-2x^{2}-16x=64$$

Combine -6x and -16x to get -22x.

$$-22x+8-2x^{2}=64$$

Subtract 8 from both sides.

$$-22x-2x^{2}=64-8$$

Subtract 8 from 64 to get 56.

$$-22x-2x^{2}=56$$

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.

$$-2x^{2}-22x=56$$

Divide both sides by -2.

$$\frac{-2x^{2}-22x}{-2}=\frac{56}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{22}{-2}\right)x=\frac{56}{-2}$$

Divide -22 by -2.

$$x^{2}+11x=\frac{56}{-2}$$

Divide 56 by -2.

$$x^{2}+11x=-28$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-28+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-28+\frac{121}{4}$$

Add -28 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{9}{4}$$

Factor x^{2}+11x+\frac{121}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{3}{2} x+\frac{11}{2}=-\frac{3}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-4 x=-7$$

Variable x cannot be equal to -4.

$$x=-7$$