Solve 1.1x=1/x+3

$$1.1 x = \frac { 1 } { x + 3 }$$

Solve for x

$$x=\frac{\sqrt{1529}}{22}-1.5\approx 0.277383163$$

$$x=-\frac{\sqrt{1529}}{22}-1.5\approx -3.277383163$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$1.1x\left(x+3\right)=1$$

Use the distributive property to multiply 1.1x by x+3.

$$1.1x^{2}+3.3x=1$$

Subtract 1 from both sides.

$$1.1x^{2}+3.3x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.1 for a, 3.3 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3.3±\sqrt{3.3^{2}-4\times 1.1\left(-1\right)}}{2\times 1.1}$$

Square 3.3 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-3.3±\sqrt{10.89-4\times 1.1\left(-1\right)}}{2\times 1.1}$$

Multiply -4 times 1.1.

$$x=\frac{-3.3±\sqrt{10.89-4.4\left(-1\right)}}{2\times 1.1}$$

Multiply -4.4 times -1.

$$x=\frac{-3.3±\sqrt{10.89+4.4}}{2\times 1.1}$$

Add 10.89 to 4.4 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3.3±\sqrt{15.29}}{2\times 1.1}$$

Take the square root of 15.29.

$$x=\frac{-3.3±\frac{\sqrt{1529}}{10}}{2\times 1.1}$$

Multiply 2 times 1.1.

$$x=\frac{-3.3±\frac{\sqrt{1529}}{10}}{2.2}$$

Now solve the equation x=\frac{-3.3±\frac{\sqrt{1529}}{10}}{2.2} when ± is plus. Add -3.3 to \frac{\sqrt{1529}}{10}.

$$x=\frac{\sqrt{1529}-33}{2.2\times 10}$$

Divide \frac{-33+\sqrt{1529}}{10} by 2.2 by multiplying \frac{-33+\sqrt{1529}}{10} by the reciprocal of 2.2.

$$x=\frac{\sqrt{1529}}{22}-\frac{3}{2}$$

Now solve the equation x=\frac{-3.3±\frac{\sqrt{1529}}{10}}{2.2} when ± is minus. Subtract \frac{\sqrt{1529}}{10} from -3.3.

$$x=\frac{-\sqrt{1529}-33}{2.2\times 10}$$

Divide \frac{-33-\sqrt{1529}}{10} by 2.2 by multiplying \frac{-33-\sqrt{1529}}{10} by the reciprocal of 2.2.

$$x=-\frac{\sqrt{1529}}{22}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1529}}{22}-\frac{3}{2} x=-\frac{\sqrt{1529}}{22}-\frac{3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$1.1x\left(x+3\right)=1$$

Use the distributive property to multiply 1.1x by x+3.

$$1.1x^{2}+3.3x=1$$

Divide both sides of the equation by 1.1, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.1x^{2}+3.3x}{1.1}=\frac{1}{1.1}$$

Dividing by 1.1 undoes the multiplication by 1.1.

$$x^{2}+\frac{3.3}{1.1}x=\frac{1}{1.1}$$

Divide 3.3 by 1.1 by multiplying 3.3 by the reciprocal of 1.1.

$$x^{2}+3x=\frac{1}{1.1}$$

Divide 1 by 1.1 by multiplying 1 by the reciprocal of 1.1.

$$x^{2}+3x=\frac{10}{11}$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{10}{11}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=\frac{10}{11}+\frac{9}{4}$$

Add \frac{10}{11} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{139}{44}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{139}{44}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{139}{44}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{1529}}{22} x+\frac{3}{2}=-\frac{\sqrt{1529}}{22}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{1529}}{22}-\frac{3}{2} x=-\frac{\sqrt{1529}}{22}-\frac{3}{2}$$