Solve x/x+2=x+3/5(x+11)

$$\frac { x } { x + 2 } = \frac { x + 3 } { 5 ( x + 11 ) }$$

Solve for x

$$x=\frac{\sqrt{649}-25}{4}\approx 0.118869601$$

$$x=\frac{-\sqrt{649}-25}{4}\approx -12.618869601$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -11,-2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x+2\right)\left(x+11\right), the least common multiple of x+2,5\left(x+11\right).

$$\left(5x+55\right)x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 5x+55 by x.

$$5x^{2}+55x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$5x^{2}+55x=x^{2}+5x+6$$

Subtract x^{2} from both sides.

$$5x^{2}+55x-x^{2}=5x+6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}+55x=5x+6$$

Subtract 5x from both sides.

$$4x^{2}+55x-5x=6$$

Combine 55x and -5x to get 50x.

$$4x^{2}+50x=6$$

Subtract 6 from both sides.

$$4x^{2}+50x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 50 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\times 4\left(-6\right)}}{2\times 4}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\times 4\left(-6\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-50±\sqrt{2500-16\left(-6\right)}}{2\times 4}$$

Multiply -16 times -6.

$$x=\frac{-50±\sqrt{2500+96}}{2\times 4}$$

Add 2500 to 96.

$$x=\frac{-50±\sqrt{2596}}{2\times 4}$$

Take the square root of 2596.

$$x=\frac{-50±2\sqrt{649}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-50±2\sqrt{649}}{8}$$

Now solve the equation x=\frac{-50±2\sqrt{649}}{8} when ± is plus. Add -50 to 2\sqrt{649}.

$$x=\frac{2\sqrt{649}-50}{8}$$

Divide -50+2\sqrt{649} by 8.

$$x=\frac{\sqrt{649}-25}{4}$$

Now solve the equation x=\frac{-50±2\sqrt{649}}{8} when ± is minus. Subtract 2\sqrt{649} from -50.

$$x=\frac{-2\sqrt{649}-50}{8}$$

Divide -50-2\sqrt{649} by 8.

$$x=\frac{-\sqrt{649}-25}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{649}-25}{4} x=\frac{-\sqrt{649}-25}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -11,-2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x+2\right)\left(x+11\right), the least common multiple of x+2,5\left(x+11\right).

$$\left(5x+55\right)x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 5x+55 by x.

$$5x^{2}+55x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$5x^{2}+55x=x^{2}+5x+6$$

Subtract x^{2} from both sides.

$$5x^{2}+55x-x^{2}=5x+6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}+55x=5x+6$$

Subtract 5x from both sides.

$$4x^{2}+55x-5x=6$$

Combine 55x and -5x to get 50x.

$$4x^{2}+50x=6$$

Divide both sides by 4.

$$\frac{4x^{2}+50x}{4}=\frac{6}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{50}{4}x=\frac{6}{4}$$

Reduce the fraction \frac{50}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{25}{2}x=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{25}{2}x=\frac{3}{2}$$

Divide \frac{25}{2}, the coefficient of the x term, by 2 to get \frac{25}{4}. Then add the square of \frac{25}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{2}x+\left(\frac{25}{4}\right)^{2}=\frac{3}{2}+\left(\frac{25}{4}\right)^{2}$$

Square \frac{25}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{25}{2}x+\frac{625}{16}=\frac{3}{2}+\frac{625}{16}$$

Add \frac{3}{2} to \frac{625}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{25}{2}x+\frac{625}{16}=\frac{649}{16}$$

Factor x^{2}+\frac{25}{2}x+\frac{625}{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}.

$$\left(x+\frac{25}{4}\right)^{2}=\frac{649}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{4}\right)^{2}}=\sqrt{\frac{649}{16}}$$

Simplify.

$$x+\frac{25}{4}=\frac{\sqrt{649}}{4} x+\frac{25}{4}=-\frac{\sqrt{649}}{4}$$

Subtract \frac{25}{4} from both sides of the equation.

$$x=\frac{\sqrt{649}-25}{4} x=\frac{-\sqrt{649}-25}{4}$$