Solve 1/x+1/x-52=1/76

$$\frac { 1 } { x } + \frac { 1 } { x - 52 } = \frac { 1 } { 76 }$$

Solve for x

$$x=2\sqrt{1613}+102\approx 182.324342512$$

$$x=102-2\sqrt{1613}\approx 21.675657488$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,52 since division by zero is not defined. Multiply both sides of the equation by 76x\left(x-52\right), the least common multiple of x,x-52,76.

$$76x-3952+76x=x\left(x-52\right)$$

Combine 76x and 76x to get 152x.

$$152x-3952=x\left(x-52\right)$$

Use the distributive property to multiply x by x-52.

$$152x-3952=x^{2}-52x$$

Subtract x^{2} from both sides.

$$152x-3952-x^{2}=-52x$$

Add 52x to both sides.

$$152x-3952-x^{2}+52x=0$$

Combine 152x and 52x to get 204x.

$$204x-3952-x^{2}=0$$

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.

$$-x^{2}+204x-3952=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 204 for b, and -3952 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-204±\sqrt{204^{2}-4\left(-1\right)\left(-3952\right)}}{2\left(-1\right)}$$

Square 204.

$$x=\frac{-204±\sqrt{41616-4\left(-1\right)\left(-3952\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-204±\sqrt{41616+4\left(-3952\right)}}{2\left(-1\right)}$$

Multiply 4 times -3952.

$$x=\frac{-204±\sqrt{41616-15808}}{2\left(-1\right)}$$

Add 41616 to -15808.

$$x=\frac{-204±\sqrt{25808}}{2\left(-1\right)}$$

Take the square root of 25808.

$$x=\frac{-204±4\sqrt{1613}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-204±4\sqrt{1613}}{-2}$$

Now solve the equation x=\frac{-204±4\sqrt{1613}}{-2} when ± is plus. Add -204 to 4\sqrt{1613}.

$$x=\frac{4\sqrt{1613}-204}{-2}$$

Divide -204+4\sqrt{1613} by -2.

$$x=102-2\sqrt{1613}$$

Now solve the equation x=\frac{-204±4\sqrt{1613}}{-2} when ± is minus. Subtract 4\sqrt{1613} from -204.

$$x=\frac{-4\sqrt{1613}-204}{-2}$$

Divide -204-4\sqrt{1613} by -2.

$$x=2\sqrt{1613}+102$$

The equation is now solved.

$$x=102-2\sqrt{1613} x=2\sqrt{1613}+102$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,52 since division by zero is not defined. Multiply both sides of the equation by 76x\left(x-52\right), the least common multiple of x,x-52,76.

$$76x-3952+76x=x\left(x-52\right)$$

Combine 76x and 76x to get 152x.

$$152x-3952=x\left(x-52\right)$$

Use the distributive property to multiply x by x-52.

$$152x-3952=x^{2}-52x$$

Subtract x^{2} from both sides.

$$152x-3952-x^{2}=-52x$$

Add 52x to both sides.

$$152x-3952-x^{2}+52x=0$$

Combine 152x and 52x to get 204x.

$$204x-3952-x^{2}=0$$

Add 3952 to both sides. Anything plus zero gives itself.

$$204x-x^{2}=3952$$

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.

$$-x^{2}+204x=3952$$

Divide both sides by -1.

$$\frac{-x^{2}+204x}{-1}=\frac{3952}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{204}{-1}x=\frac{3952}{-1}$$

Divide 204 by -1.

$$x^{2}-204x=\frac{3952}{-1}$$

Divide 3952 by -1.

$$x^{2}-204x=-3952$$

Divide -204, the coefficient of the x term, by 2 to get -102. Then add the square of -102 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-204x+\left(-102\right)^{2}=-3952+\left(-102\right)^{2}$$

Square -102.

$$x^{2}-204x+10404=-3952+10404$$

Add -3952 to 10404.

$$x^{2}-204x+10404=6452$$

Factor x^{2}-204x+10404. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-102\right)^{2}=6452$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-102\right)^{2}}=\sqrt{6452}$$

Simplify.

$$x-102=2\sqrt{1613} x-102=-2\sqrt{1613}$$

Add 102 to both sides of the equation.

$$x=2\sqrt{1613}+102 x=102-2\sqrt{1613}$$