Solve 1/x+40+1/x=1/48

$$\frac { 1 } { x + 40 } + \frac { 1 } { x } = \frac { 1 } { 48 }$$

Solve for x

$$x=-24$$

$$x=80$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -40,0 since division by zero is not defined. Multiply both sides of the equation by 48x\left(x+40\right), the least common multiple of x+40,x,48.

$$48x+48x+1920=x\left(x+40\right)$$

Combine 48x and 48x to get 96x.

$$96x+1920=x\left(x+40\right)$$

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

$$96x+1920=x^{2}+40x$$

Subtract x^{2} from both sides.

$$96x+1920-x^{2}=40x$$

Subtract 40x from both sides.

$$96x+1920-x^{2}-40x=0$$

Combine 96x and -40x to get 56x.

$$56x+1920-x^{2}=0$$

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

$$-x^{2}+56x+1920=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+1920. To find a and b, set up a system to be solved.

$$a+b=56 ab=-1920=-1920$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1920.

$$-1,1920 -2,960 -3,640 -4,480 -5,384 -6,320 -8,240 -10,192 -12,160 -15,128 -16,120 -20,96 -24,80 -30,64 -32,60 -40,48$$

Calculate the sum for each pair.

$$-1+1920=1919 -2+960=958 -3+640=637 -4+480=476 -5+384=379 -6+320=314 -8+240=232 -10+192=182 -12+160=148 -15+128=113 -16+120=104 -20+96=76 -24+80=56 -30+64=34 -32+60=28 -40+48=8$$

The solution is the pair that gives sum 56.

$$a=80 b=-24$$

Rewrite -x^{2}+56x+1920 as \left(-x^{2}+80x\right)+\left(-24x+1920\right).

$$\left(-x^{2}+80x\right)+\left(-24x+1920\right)$$

Factor out -x in the first and -24 in the second group.

$$-x\left(x-80\right)-24\left(x-80\right)$$

Factor out common term x-80 by using distributive property.

$$\left(x-80\right)\left(-x-24\right)$$

To find equation solutions, solve x-80=0 and -x-24=0.

$$x=80 x=-24$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -40,0 since division by zero is not defined. Multiply both sides of the equation by 48x\left(x+40\right), the least common multiple of x+40,x,48.

$$48x+48x+1920=x\left(x+40\right)$$

Combine 48x and 48x to get 96x.

$$96x+1920=x\left(x+40\right)$$

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

$$96x+1920=x^{2}+40x$$

Subtract x^{2} from both sides.

$$96x+1920-x^{2}=40x$$

Subtract 40x from both sides.

$$96x+1920-x^{2}-40x=0$$

Combine 96x and -40x to get 56x.

$$56x+1920-x^{2}=0$$

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.

$$-x^{2}+56x+1920=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 56 for b, and 1920 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\left(-1\right)\times 1920}}{2\left(-1\right)}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\left(-1\right)\times 1920}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-56±\sqrt{3136+4\times 1920}}{2\left(-1\right)}$$

Multiply 4 times 1920.

$$x=\frac{-56±\sqrt{3136+7680}}{2\left(-1\right)}$$

Add 3136 to 7680.

$$x=\frac{-56±\sqrt{10816}}{2\left(-1\right)}$$

Take the square root of 10816.

$$x=\frac{-56±104}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-56±104}{-2}$$

Now solve the equation x=\frac{-56±104}{-2} when ± is plus. Add -56 to 104.

$$x=\frac{48}{-2}$$

Divide 48 by -2.

$$x=-24$$

Now solve the equation x=\frac{-56±104}{-2} when ± is minus. Subtract 104 from -56.

$$x=-\frac{160}{-2}$$

Divide -160 by -2.

$$x=80$$

The equation is now solved.

$$x=-24 x=80$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -40,0 since division by zero is not defined. Multiply both sides of the equation by 48x\left(x+40\right), the least common multiple of x+40,x,48.

$$48x+48x+1920=x\left(x+40\right)$$

Combine 48x and 48x to get 96x.

$$96x+1920=x\left(x+40\right)$$

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

$$96x+1920=x^{2}+40x$$

Subtract x^{2} from both sides.

$$96x+1920-x^{2}=40x$$

Subtract 40x from both sides.

$$96x+1920-x^{2}-40x=0$$

Combine 96x and -40x to get 56x.

$$56x+1920-x^{2}=0$$

Subtract 1920 from both sides. Anything subtracted from zero gives its negation.

$$56x-x^{2}=-1920$$

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.

$$-x^{2}+56x=-1920$$

Divide both sides by -1.

$$\frac{-x^{2}+56x}{-1}=-\frac{1920}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{56}{-1}x=-\frac{1920}{-1}$$

Divide 56 by -1.

$$x^{2}-56x=-\frac{1920}{-1}$$

Divide -1920 by -1.

$$x^{2}-56x=1920$$

Divide -56, the coefficient of the x term, by 2 to get -28. Then add the square of -28 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-56x+\left(-28\right)^{2}=1920+\left(-28\right)^{2}$$

Square -28.

$$x^{2}-56x+784=1920+784$$

Add 1920 to 784.

$$x^{2}-56x+784=2704$$

Factor x^{2}-56x+784. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-28\right)^{2}=2704$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-28\right)^{2}}=\sqrt{2704}$$

Simplify.

$$x-28=52 x-28=-52$$

Add 28 to both sides of the equation.

$$x=80 x=-24$$