Solve 50/x+18+8/18-x=3

$$\frac{ 50 }{ x+18 } + \frac{ 8 }{ 18-x } =3$$

Solve for x

$$x=2$$

$$x=12$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -18,18 since division by zero is not defined. Multiply both sides of the equation by \left(x-18\right)\left(x+18\right), the least common multiple of x+18,18-x.

$$\left(x-18\right)\times 50-\left(18+x\right)\times 8=3\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply x-18 by 50.

$$50x-900-\left(18+x\right)\times 8=3\left(x-18\right)\left(x+18\right)$$

Multiply -1 and 8 to get -8.

$$50x-900-8\left(18+x\right)=3\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply -8 by 18+x.

$$50x-900-144-8x=3\left(x-18\right)\left(x+18\right)$$

Subtract 144 from -900 to get -1044.

$$50x-1044-8x=3\left(x-18\right)\left(x+18\right)$$

Combine 50x and -8x to get 42x.

$$42x-1044=3\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply 3 by x-18.

$$42x-1044=\left(3x-54\right)\left(x+18\right)$$

Use the distributive property to multiply 3x-54 by x+18 and combine like terms.

$$42x-1044=3x^{2}-972$$

Subtract 3x^{2} from both sides.

$$42x-1044-3x^{2}=-972$$

Add 972 to both sides.

$$42x-1044-3x^{2}+972=0$$

Add -1044 and 972 to get -72.

$$42x-72-3x^{2}=0$$

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.

$$-3x^{2}+42x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 42 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-42±\sqrt{42^{2}-4\left(-3\right)\left(-72\right)}}{2\left(-3\right)}$$

Square 42.

$$x=\frac{-42±\sqrt{1764-4\left(-3\right)\left(-72\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-42±\sqrt{1764+12\left(-72\right)}}{2\left(-3\right)}$$

Multiply 12 times -72.

$$x=\frac{-42±\sqrt{1764-864}}{2\left(-3\right)}$$

Add 1764 to -864.

$$x=\frac{-42±\sqrt{900}}{2\left(-3\right)}$$

Take the square root of 900.

$$x=\frac{-42±30}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-42±30}{-6}$$

Now solve the equation x=\frac{-42±30}{-6} when ± is plus. Add -42 to 30.

$$x=-\frac{12}{-6}$$

Divide -12 by -6.

$$x=2$$

Now solve the equation x=\frac{-42±30}{-6} when ± is minus. Subtract 30 from -42.

$$x=-\frac{72}{-6}$$

Divide -72 by -6.

$$x=12$$

The equation is now solved.

$$x=2 x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -18,18 since division by zero is not defined. Multiply both sides of the equation by \left(x-18\right)\left(x+18\right), the least common multiple of x+18,18-x.

$$\left(x-18\right)\times 50-\left(18+x\right)\times 8=3\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply x-18 by 50.

$$50x-900-\left(18+x\right)\times 8=3\left(x-18\right)\left(x+18\right)$$

Multiply -1 and 8 to get -8.

$$50x-900-8\left(18+x\right)=3\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply -8 by 18+x.

$$50x-900-144-8x=3\left(x-18\right)\left(x+18\right)$$

Subtract 144 from -900 to get -1044.

$$50x-1044-8x=3\left(x-18\right)\left(x+18\right)$$

Combine 50x and -8x to get 42x.

$$42x-1044=3\left(x-18\right)\left(x+18\right)$$

Use the distributive property to multiply 3 by x-18.

$$42x-1044=\left(3x-54\right)\left(x+18\right)$$

Use the distributive property to multiply 3x-54 by x+18 and combine like terms.

$$42x-1044=3x^{2}-972$$

Subtract 3x^{2} from both sides.

$$42x-1044-3x^{2}=-972$$

Add 1044 to both sides.

$$42x-3x^{2}=-972+1044$$

Add -972 and 1044 to get 72.

$$42x-3x^{2}=72$$

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.

$$-3x^{2}+42x=72$$

Divide both sides by -3.

$$\frac{-3x^{2}+42x}{-3}=\frac{72}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{42}{-3}x=\frac{72}{-3}$$

Divide 42 by -3.

$$x^{2}-14x=\frac{72}{-3}$$

Divide 72 by -3.

$$x^{2}-14x=-24$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-24+49$$

Add -24 to 49.

$$x^{2}-14x+49=25$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-7=5 x-7=-5$$

Add 7 to both sides of the equation.

$$x=12 x=2$$