Solve 1/x+12-3x+40/4-(x+10)^2=x/4x+32+3/x+8

$$\frac { 1 } { x + 12 } - \frac { 3 x + 40 } { 4 - ( x + 10 ) ^ { 2 } } = \frac { x } { 4 x + 32 } + \frac { 3 } { x + 8 }$$

Solve for x

$$x=4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -12,-8 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+8\right)\left(x+12\right), the least common multiple of x+12,4-\left(x+10\right)^{2},4x+32,x+8.

$$4x+32+4\left(3x+40\right)=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply 4 by 3x+40.

$$4x+32+12x+160=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Combine 4x and 12x to get 16x.

$$16x+32+160=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Add 32 and 160 to get 192.

$$16x+192=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply x+12 by x.

$$16x+192=x^{2}+12x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply 4x+48 by 3.

$$16x+192=x^{2}+12x+12x+144$$

Combine 12x and 12x to get 24x.

$$16x+192=x^{2}+24x+144$$

Subtract x^{2} from both sides.

$$16x+192-x^{2}=24x+144$$

Subtract 24x from both sides.

$$16x+192-x^{2}-24x=144$$

Combine 16x and -24x to get -8x.

$$-8x+192-x^{2}=144$$

Subtract 144 from both sides.

$$-8x+192-x^{2}-144=0$$

Subtract 144 from 192 to get 48.

$$-8x+48-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-8x+48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+48. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-48=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -8.

$$a=4 b=-12$$

Rewrite -x^{2}-8x+48 as \left(-x^{2}+4x\right)+\left(-12x+48\right).

$$\left(-x^{2}+4x\right)+\left(-12x+48\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(-x+4\right)+12\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(x+12\right)$$

To find equation solutions, solve -x+4=0 and x+12=0.

$$x=4 x=-12$$

Variable x cannot be equal to -12.

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -12,-8 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+8\right)\left(x+12\right), the least common multiple of x+12,4-\left(x+10\right)^{2},4x+32,x+8.

$$4x+32+4\left(3x+40\right)=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply 4 by 3x+40.

$$4x+32+12x+160=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Combine 4x and 12x to get 16x.

$$16x+32+160=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Add 32 and 160 to get 192.

$$16x+192=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply x+12 by x.

$$16x+192=x^{2}+12x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply 4x+48 by 3.

$$16x+192=x^{2}+12x+12x+144$$

Combine 12x and 12x to get 24x.

$$16x+192=x^{2}+24x+144$$

Subtract x^{2} from both sides.

$$16x+192-x^{2}=24x+144$$

Subtract 24x from both sides.

$$16x+192-x^{2}-24x=144$$

Combine 16x and -24x to get -8x.

$$-8x+192-x^{2}=144$$

Subtract 144 from both sides.

$$-8x+192-x^{2}-144=0$$

Subtract 144 from 192 to get 48.

$$-8x+48-x^{2}=0$$

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.

$$-x^{2}-8x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 48}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 48}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\times 48}}{2\left(-1\right)}$$

Multiply 4 times 48.

$$x=\frac{-\left(-8\right)±\sqrt{64+192}}{2\left(-1\right)}$$

Add 64 to 192.

$$x=\frac{-\left(-8\right)±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

$$x=\frac{-\left(-8\right)±16}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±16}{-2}$$

Now solve the equation x=\frac{8±16}{-2} when ± is plus. Add 8 to 16.

$$x=\frac{24}{-2}$$

Divide 24 by -2.

$$x=-12$$

Now solve the equation x=\frac{8±16}{-2} when ± is minus. Subtract 16 from 8.

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

Divide -8 by -2.

$$x=4$$

The equation is now solved.

$$x=-12 x=4$$

Variable x cannot be equal to -12.

$$x=4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -12,-8 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+8\right)\left(x+12\right), the least common multiple of x+12,4-\left(x+10\right)^{2},4x+32,x+8.

$$4x+32+4\left(3x+40\right)=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply 4 by 3x+40.

$$4x+32+12x+160=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Combine 4x and 12x to get 16x.

$$16x+32+160=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Add 32 and 160 to get 192.

$$16x+192=\left(x+12\right)x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply x+12 by x.

$$16x+192=x^{2}+12x+\left(4x+48\right)\times 3$$

Use the distributive property to multiply 4x+48 by 3.

$$16x+192=x^{2}+12x+12x+144$$

Combine 12x and 12x to get 24x.

$$16x+192=x^{2}+24x+144$$

Subtract x^{2} from both sides.

$$16x+192-x^{2}=24x+144$$

Subtract 24x from both sides.

$$16x+192-x^{2}-24x=144$$

Combine 16x and -24x to get -8x.

$$-8x+192-x^{2}=144$$

Subtract 192 from both sides.

$$-8x-x^{2}=144-192$$

Subtract 192 from 144 to get -48.

$$-8x-x^{2}=-48$$

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.

$$-x^{2}-8x=-48$$

Divide both sides by -1.

$$\frac{-x^{2}-8x}{-1}=-\frac{48}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=-\frac{48}{-1}$$

Divide -8 by -1.

$$x^{2}+8x=-\frac{48}{-1}$$

Divide -48 by -1.

$$x^{2}+8x=48$$

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.

$$x^{2}+8x+4^{2}=48+4^{2}$$

Square 4.

$$x^{2}+8x+16=48+16$$

Add 48 to 16.

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

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}.

$$\left(x+4\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{64}$$

Simplify.

$$x+4=8 x+4=-8$$

Subtract 4 from both sides of the equation.

$$x=4 x=-12$$

Variable x cannot be equal to -12.

$$x=4$$