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

$$x=-1$$

Steps Using the Quadratic Formula

Express 2\times \frac{-x+1}{-x+4} as a single fraction.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{-x+4}{-x+4}.

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

Since \frac{4\left(-x+4\right)}{-x+4} and \frac{2\left(-x\right)+2}{-x+4} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4\left(-x+4\right)-\left(2\left(-x\right)+2\right).

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

Combine like terms in -4x+16+2x-2.

$$\frac{-2x+14}{-x+4}\left(-x+4\right)=16$$

Express \frac{-2x+14}{-x+4}\left(-x+4\right) as a single fraction.

$$\frac{\left(-2x+14\right)\left(-x+4\right)}{-x+4}=16$$

Use the distributive property to multiply -2x+14 by -x+4.

$$\frac{-2x\left(-x\right)-8x+14\left(-x\right)+56}{-x+4}=16$$

Multiply -2 and -1 to get 2.

$$\frac{2xx-8x+14\left(-x\right)+56}{-x+4}=16$$

Multiply x and x to get x^{2}.

$$\frac{2x^{2}-8x+14\left(-x\right)+56}{-x+4}=16$$

Subtract 16 from both sides.

$$\frac{2x^{2}-8x+14\left(-x\right)+56}{-x+4}-16=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{-x+4}{-x+4}.

$$\frac{2x^{2}-8x+14\left(-x\right)+56}{-x+4}-\frac{16\left(-x+4\right)}{-x+4}=0$$

Since \frac{2x^{2}-8x+14\left(-x\right)+56}{-x+4} and \frac{16\left(-x+4\right)}{-x+4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}-8x+14\left(-x\right)+56-16\left(-x+4\right)}{-x+4}=0$$

Do the multiplications in 2x^{2}-8x+14\left(-x\right)+56-16\left(-x+4\right).

$$\frac{2x^{2}-8x-14x+56+16x-64}{-x+4}=0$$

Combine like terms in 2x^{2}-8x-14x+56+16x-64.

$$\frac{2x^{2}-6x-8}{-x+4}=0$$

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

$$2x^{2}-6x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -6 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-8\right)}}{2\times 2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-8\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\left(-8\right)}}{2\times 2}$$

Multiply -8 times -8.

$$x=\frac{-\left(-6\right)±\sqrt{36+64}}{2\times 2}$$

Add 36 to 64.

$$x=\frac{-\left(-6\right)±\sqrt{100}}{2\times 2}$$

Take the square root of 100.

$$x=\frac{-\left(-6\right)±10}{2\times 2}$$

The opposite of -6 is 6.

$$x=\frac{6±10}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{6±10}{4}$$

Now solve the equation x=\frac{6±10}{4} when ± is plus. Add 6 to 10.

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

Divide 16 by 4.

$$x=4$$

Now solve the equation x=\frac{6±10}{4} when ± is minus. Subtract 10 from 6.

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

Divide -4 by 4.

$$x=-1$$

The equation is now solved.

$$x=4 x=-1$$

Variable x cannot be equal to 4.

$$x=-1$$

Steps for Completing the Square

Express 2\times \frac{-x+1}{-x+4} as a single fraction.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{-x+4}{-x+4}.

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

Since \frac{4\left(-x+4\right)}{-x+4} and \frac{2\left(-x\right)+2}{-x+4} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4\left(-x+4\right)-\left(2\left(-x\right)+2\right).

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

Combine like terms in -4x+16+2x-2.

$$\frac{-2x+14}{-x+4}\left(-x+4\right)=16$$

Express \frac{-2x+14}{-x+4}\left(-x+4\right) as a single fraction.

$$\frac{\left(-2x+14\right)\left(-x+4\right)}{-x+4}=16$$

Use the distributive property to multiply -2x+14 by -x+4.

$$\frac{-2x\left(-x\right)-8x+14\left(-x\right)+56}{-x+4}=16$$

Multiply -2 and -1 to get 2.

$$\frac{2xx-8x+14\left(-x\right)+56}{-x+4}=16$$

Multiply x and x to get x^{2}.

$$\frac{2x^{2}-8x+14\left(-x\right)+56}{-x+4}=16$$

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

$$2x^{2}-8x+14\left(-1\right)x+56=16\left(-x+4\right)$$

Multiply 14 and -1 to get -14.

$$2x^{2}-8x-14x+56=16\left(-x+4\right)$$

Combine -8x and -14x to get -22x.

$$2x^{2}-22x+56=16\left(-x+4\right)$$

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

$$2x^{2}-22x+56=-16x+64$$

Add 16x to both sides.

$$2x^{2}-22x+56+16x=64$$

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

$$2x^{2}-6x+56=64$$

Subtract 56 from both sides.

$$2x^{2}-6x=64-56$$

Subtract 56 from 64 to get 8.

$$2x^{2}-6x=8$$

Divide both sides by 2.

$$\frac{2x^{2}-6x}{2}=\frac{8}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{6}{2}\right)x=\frac{8}{2}$$

Divide -6 by 2.

$$x^{2}-3x=\frac{8}{2}$$

Divide 8 by 2.

$$x^{2}-3x=4$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=4+\frac{9}{4}$$

Add 4 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{25}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{5}{2} x-\frac{3}{2}=-\frac{5}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=4 x=-1$$

Variable x cannot be equal to 4.

$$x=-1$$