Solve 57/x+2-21/x+6=1

$$\frac { 57 } { x + 2 } - \frac { 21 } { x + 6 } = 1$$

Solve for x

$$x=-8$$

$$x=36$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+6\right), the least common multiple of x+2,x+6.

$$\left(x+6\right)\times 57-\left(x+2\right)\times 21=\left(x+2\right)\left(x+6\right)$$

Use the distributive property to multiply x+6 by 57.

$$57x+342-\left(x+2\right)\times 21=\left(x+2\right)\left(x+6\right)$$

Use the distributive property to multiply x+2 by 21.

$$57x+342-\left(21x+42\right)=\left(x+2\right)\left(x+6\right)$$

To find the opposite of 21x+42, find the opposite of each term.

$$57x+342-21x-42=\left(x+2\right)\left(x+6\right)$$

Combine 57x and -21x to get 36x.

$$36x+342-42=\left(x+2\right)\left(x+6\right)$$

Subtract 42 from 342 to get 300.

$$36x+300=\left(x+2\right)\left(x+6\right)$$

Use the distributive property to multiply x+2 by x+6 and combine like terms.

$$36x+300=x^{2}+8x+12$$

Subtract x^{2} from both sides.

$$36x+300-x^{2}=8x+12$$

Subtract 8x from both sides.

$$36x+300-x^{2}-8x=12$$

Combine 36x and -8x to get 28x.

$$28x+300-x^{2}=12$$

Subtract 12 from both sides.

$$28x+300-x^{2}-12=0$$

Subtract 12 from 300 to get 288.

$$28x+288-x^{2}=0$$

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.

$$-x^{2}+28x+288=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 28 for b, and 288 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-1\right)\times 288}}{2\left(-1\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-1\right)\times 288}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-28±\sqrt{784+4\times 288}}{2\left(-1\right)}$$

Multiply 4 times 288.

$$x=\frac{-28±\sqrt{784+1152}}{2\left(-1\right)}$$

Add 784 to 1152.

$$x=\frac{-28±\sqrt{1936}}{2\left(-1\right)}$$

Take the square root of 1936.

$$x=\frac{-28±44}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-28±44}{-2}$$

Now solve the equation x=\frac{-28±44}{-2} when ± is plus. Add -28 to 44.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{-28±44}{-2} when ± is minus. Subtract 44 from -28.

$$x=-\frac{72}{-2}$$

Divide -72 by -2.

$$x=36$$

The equation is now solved.

$$x=-8 x=36$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+6\right), the least common multiple of x+2,x+6.

$$\left(x+6\right)\times 57-\left(x+2\right)\times 21=\left(x+2\right)\left(x+6\right)$$

Use the distributive property to multiply x+6 by 57.

$$57x+342-\left(x+2\right)\times 21=\left(x+2\right)\left(x+6\right)$$

Use the distributive property to multiply x+2 by 21.

$$57x+342-\left(21x+42\right)=\left(x+2\right)\left(x+6\right)$$

To find the opposite of 21x+42, find the opposite of each term.

$$57x+342-21x-42=\left(x+2\right)\left(x+6\right)$$

Combine 57x and -21x to get 36x.

$$36x+342-42=\left(x+2\right)\left(x+6\right)$$

Subtract 42 from 342 to get 300.

$$36x+300=\left(x+2\right)\left(x+6\right)$$

Use the distributive property to multiply x+2 by x+6 and combine like terms.

$$36x+300=x^{2}+8x+12$$

Subtract x^{2} from both sides.

$$36x+300-x^{2}=8x+12$$

Subtract 8x from both sides.

$$36x+300-x^{2}-8x=12$$

Combine 36x and -8x to get 28x.

$$28x+300-x^{2}=12$$

Subtract 300 from both sides.

$$28x-x^{2}=12-300$$

Subtract 300 from 12 to get -288.

$$28x-x^{2}=-288$$

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.

$$-x^{2}+28x=-288$$

Divide both sides by -1.

$$\frac{-x^{2}+28x}{-1}=-\frac{288}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{28}{-1}x=-\frac{288}{-1}$$

Divide 28 by -1.

$$x^{2}-28x=-\frac{288}{-1}$$

Divide -288 by -1.

$$x^{2}-28x=288$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-28x+\left(-14\right)^{2}=288+\left(-14\right)^{2}$$

Square -14.

$$x^{2}-28x+196=288+196$$

Add 288 to 196.

$$x^{2}-28x+196=484$$

Factor x^{2}-28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-14\right)^{2}=484$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{484}$$

Simplify.

$$x-14=22 x-14=-22$$

Add 14 to both sides of the equation.

$$x=36 x=-8$$