Solve (1/8+7/x+6)*4+1/x+6(x-4)=1

$$( \frac { 1 } { 8 } + \frac { 7 } { x + 6 } ) \times 4 + \frac { 1 } { x + 6 } ( x - 4 ) = 1$$

Solve for x

$$x=-42$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 8\left(x+6\right), the least common multiple of 8,x+6.

$$\left(\frac{1}{8}+\frac{7}{x+6}\right)\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8 and x+6 is 8\left(x+6\right). Multiply \frac{1}{8} times \frac{x+6}{x+6}. Multiply \frac{7}{x+6} times \frac{8}{8}.

$$\left(\frac{x+6}{8\left(x+6\right)}+\frac{7\times 8}{8\left(x+6\right)}\right)\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Since \frac{x+6}{8\left(x+6\right)} and \frac{7\times 8}{8\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+6+7\times 8}{8\left(x+6\right)}\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Do the multiplications in x+6+7\times 8.

$$\frac{x+6+56}{8\left(x+6\right)}\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Combine like terms in x+6+56.

$$\frac{x+62}{8\left(x+6\right)}\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Multiply 4 and 8 to get 32.

$$\frac{x+62}{8\left(x+6\right)}\times 32\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Express \frac{x+62}{8\left(x+6\right)}\times 32 as a single fraction.

$$\frac{\left(x+62\right)\times 32}{8\left(x+6\right)}\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Cancel out 8 in both numerator and denominator.

$$\frac{4\left(x+62\right)}{x+6}\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Express \frac{4\left(x+62\right)}{x+6}\left(x+6\right) as a single fraction.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Multiply 8 and 1 to get 8.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+8\left(x-4\right)=8\left(x+6\right)$$

Use the distributive property to multiply 8 by x-4.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+8x-32=8\left(x+6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8x-32 times \frac{x+6}{x+6}.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+\frac{\left(8x-32\right)\left(x+6\right)}{x+6}=8\left(x+6\right)$$

Since \frac{4\left(x+62\right)\left(x+6\right)}{x+6} and \frac{\left(8x-32\right)\left(x+6\right)}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x+62\right)\left(x+6\right)+\left(8x-32\right)\left(x+6\right)}{x+6}=8\left(x+6\right)$$

Do the multiplications in 4\left(x+62\right)\left(x+6\right)+\left(8x-32\right)\left(x+6\right).

$$\frac{4x^{2}+24x+248x+1488+8x^{2}+48x-32x-192}{x+6}=8\left(x+6\right)$$

Combine like terms in 4x^{2}+24x+248x+1488+8x^{2}+48x-32x-192.

$$\frac{12x^{2}+288x+1296}{x+6}=8\left(x+6\right)$$

Use the distributive property to multiply 8 by x+6.

$$\frac{12x^{2}+288x+1296}{x+6}=8x+48$$

Subtract 8x from both sides.

$$\frac{12x^{2}+288x+1296}{x+6}-8x=48$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -8x times \frac{x+6}{x+6}.

$$\frac{12x^{2}+288x+1296}{x+6}+\frac{-8x\left(x+6\right)}{x+6}=48$$

Since \frac{12x^{2}+288x+1296}{x+6} and \frac{-8x\left(x+6\right)}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{12x^{2}+288x+1296-8x\left(x+6\right)}{x+6}=48$$

Do the multiplications in 12x^{2}+288x+1296-8x\left(x+6\right).

$$\frac{12x^{2}+288x+1296-8x^{2}-48x}{x+6}=48$$

Combine like terms in 12x^{2}+288x+1296-8x^{2}-48x.

$$\frac{4x^{2}+240x+1296}{x+6}=48$$

Subtract 48 from both sides.

$$\frac{4x^{2}+240x+1296}{x+6}-48=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 48 times \frac{x+6}{x+6}.

$$\frac{4x^{2}+240x+1296}{x+6}-\frac{48\left(x+6\right)}{x+6}=0$$

Since \frac{4x^{2}+240x+1296}{x+6} and \frac{48\left(x+6\right)}{x+6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x^{2}+240x+1296-48\left(x+6\right)}{x+6}=0$$

Do the multiplications in 4x^{2}+240x+1296-48\left(x+6\right).

$$\frac{4x^{2}+240x+1296-48x-288}{x+6}=0$$

Combine like terms in 4x^{2}+240x+1296-48x-288.

$$\frac{4x^{2}+192x+1008}{x+6}=0$$

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by x+6.

$$4x^{2}+192x+1008=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 192 for b, and 1008 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-192±\sqrt{192^{2}-4\times 4\times 1008}}{2\times 4}$$

Square 192.

$$x=\frac{-192±\sqrt{36864-4\times 4\times 1008}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-192±\sqrt{36864-16\times 1008}}{2\times 4}$$

Multiply -16 times 1008.

$$x=\frac{-192±\sqrt{36864-16128}}{2\times 4}$$

Add 36864 to -16128.

$$x=\frac{-192±\sqrt{20736}}{2\times 4}$$

Take the square root of 20736.

$$x=\frac{-192±144}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-192±144}{8}$$

Now solve the equation x=\frac{-192±144}{8} when ± is plus. Add -192 to 144.

$$x=-\frac{48}{8}$$

Divide -48 by 8.

$$x=-6$$

Now solve the equation x=\frac{-192±144}{8} when ± is minus. Subtract 144 from -192.

$$x=-\frac{336}{8}$$

Divide -336 by 8.

$$x=-42$$

The equation is now solved.

$$x=-6 x=-42$$

Variable x cannot be equal to -6.

$$x=-42$$

Steps for Completing the Square

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 8\left(x+6\right), the least common multiple of 8,x+6.

$$\left(\frac{1}{8}+\frac{7}{x+6}\right)\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8 and x+6 is 8\left(x+6\right). Multiply \frac{1}{8} times \frac{x+6}{x+6}. Multiply \frac{7}{x+6} times \frac{8}{8}.

$$\left(\frac{x+6}{8\left(x+6\right)}+\frac{7\times 8}{8\left(x+6\right)}\right)\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Since \frac{x+6}{8\left(x+6\right)} and \frac{7\times 8}{8\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+6+7\times 8}{8\left(x+6\right)}\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Do the multiplications in x+6+7\times 8.

$$\frac{x+6+56}{8\left(x+6\right)}\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Combine like terms in x+6+56.

$$\frac{x+62}{8\left(x+6\right)}\times 4\times 8\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Multiply 4 and 8 to get 32.

$$\frac{x+62}{8\left(x+6\right)}\times 32\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Express \frac{x+62}{8\left(x+6\right)}\times 32 as a single fraction.

$$\frac{\left(x+62\right)\times 32}{8\left(x+6\right)}\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Cancel out 8 in both numerator and denominator.

$$\frac{4\left(x+62\right)}{x+6}\left(x+6\right)+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Express \frac{4\left(x+62\right)}{x+6}\left(x+6\right) as a single fraction.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+8\times 1\left(x-4\right)=8\left(x+6\right)$$

Multiply 8 and 1 to get 8.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+8\left(x-4\right)=8\left(x+6\right)$$

Use the distributive property to multiply 8 by x-4.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+8x-32=8\left(x+6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8x-32 times \frac{x+6}{x+6}.

$$\frac{4\left(x+62\right)\left(x+6\right)}{x+6}+\frac{\left(8x-32\right)\left(x+6\right)}{x+6}=8\left(x+6\right)$$

Since \frac{4\left(x+62\right)\left(x+6\right)}{x+6} and \frac{\left(8x-32\right)\left(x+6\right)}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x+62\right)\left(x+6\right)+\left(8x-32\right)\left(x+6\right)}{x+6}=8\left(x+6\right)$$

Do the multiplications in 4\left(x+62\right)\left(x+6\right)+\left(8x-32\right)\left(x+6\right).

$$\frac{4x^{2}+24x+248x+1488+8x^{2}+48x-32x-192}{x+6}=8\left(x+6\right)$$

Combine like terms in 4x^{2}+24x+248x+1488+8x^{2}+48x-32x-192.

$$\frac{12x^{2}+288x+1296}{x+6}=8\left(x+6\right)$$

Use the distributive property to multiply 8 by x+6.

$$\frac{12x^{2}+288x+1296}{x+6}=8x+48$$

Subtract 8x from both sides.

$$\frac{12x^{2}+288x+1296}{x+6}-8x=48$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -8x times \frac{x+6}{x+6}.

$$\frac{12x^{2}+288x+1296}{x+6}+\frac{-8x\left(x+6\right)}{x+6}=48$$

Since \frac{12x^{2}+288x+1296}{x+6} and \frac{-8x\left(x+6\right)}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{12x^{2}+288x+1296-8x\left(x+6\right)}{x+6}=48$$

Do the multiplications in 12x^{2}+288x+1296-8x\left(x+6\right).

$$\frac{12x^{2}+288x+1296-8x^{2}-48x}{x+6}=48$$

Combine like terms in 12x^{2}+288x+1296-8x^{2}-48x.

$$\frac{4x^{2}+240x+1296}{x+6}=48$$

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by x+6.

$$4x^{2}+240x+1296=48\left(x+6\right)$$

Use the distributive property to multiply 48 by x+6.

$$4x^{2}+240x+1296=48x+288$$

Subtract 48x from both sides.

$$4x^{2}+240x+1296-48x=288$$

Combine 240x and -48x to get 192x.

$$4x^{2}+192x+1296=288$$

Subtract 1296 from both sides.

$$4x^{2}+192x=288-1296$$

Subtract 1296 from 288 to get -1008.

$$4x^{2}+192x=-1008$$

Divide both sides by 4.

$$\frac{4x^{2}+192x}{4}=-\frac{1008}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{192}{4}x=-\frac{1008}{4}$$

Divide 192 by 4.

$$x^{2}+48x=-\frac{1008}{4}$$

Divide -1008 by 4.

$$x^{2}+48x=-252$$

Divide 48, the coefficient of the x term, by 2 to get 24. Then add the square of 24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+48x+24^{2}=-252+24^{2}$$

Square 24.

$$x^{2}+48x+576=-252+576$$

Add -252 to 576.

$$x^{2}+48x+576=324$$

Factor x^{2}+48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+24\right)^{2}=324$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{324}$$

Simplify.

$$x+24=18 x+24=-18$$

Subtract 24 from both sides of the equation.

$$x=-6 x=-42$$

Variable x cannot be equal to -6.

$$x=-42$$