Solve 5x-3/x=1-x/x+2

$$\frac { 5 x - 3 } { x } = \frac { 1 - x } { x + 2 }$$

Solve for x

$$x=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

$$x=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\left(5x-3\right)=x\left(1-x\right)$$

Use the distributive property to multiply x+2 by 5x-3 and combine like terms.

$$5x^{2}+7x-6=x\left(1-x\right)$$

Use the distributive property to multiply x by 1-x.

$$5x^{2}+7x-6=x-x^{2}$$

Subtract x from both sides.

$$5x^{2}+7x-6-x=-x^{2}$$

Combine 7x and -x to get 6x.

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

Add x^{2} to both sides.

$$5x^{2}+6x-6+x^{2}=0$$

Combine 5x^{2} and x^{2} to get 6x^{2}.

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

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

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

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 6\left(-6\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-6±\sqrt{36-24\left(-6\right)}}{2\times 6}$$

Multiply -24 times -6.

$$x=\frac{-6±\sqrt{36+144}}{2\times 6}$$

Add 36 to 144.

$$x=\frac{-6±\sqrt{180}}{2\times 6}$$

Take the square root of 180.

$$x=\frac{-6±6\sqrt{5}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-6±6\sqrt{5}}{12}$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{12} when ± is plus. Add -6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}-6}{12}$$

Divide -6+6\sqrt{5} by 12.

$$x=\frac{\sqrt{5}-1}{2}$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{12} when ± is minus. Subtract 6\sqrt{5} from -6.

$$x=\frac{-6\sqrt{5}-6}{12}$$

Divide -6-6\sqrt{5} by 12.

$$x=\frac{-\sqrt{5}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{5}-1}{2} x=\frac{-\sqrt{5}-1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\left(5x-3\right)=x\left(1-x\right)$$

Use the distributive property to multiply x+2 by 5x-3 and combine like terms.

$$5x^{2}+7x-6=x\left(1-x\right)$$

Use the distributive property to multiply x by 1-x.

$$5x^{2}+7x-6=x-x^{2}$$

Subtract x from both sides.

$$5x^{2}+7x-6-x=-x^{2}$$

Combine 7x and -x to get 6x.

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

Add x^{2} to both sides.

$$5x^{2}+6x-6+x^{2}=0$$

Combine 5x^{2} and x^{2} to get 6x^{2}.

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

Add 6 to both sides. Anything plus zero gives itself.

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

Divide both sides by 6.

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

Dividing by 6 undoes the multiplication by 6.

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

Divide 6 by 6.

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

Divide 6 by 6.

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

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=1+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=1+\frac{1}{4}$$

Add 1 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{5}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{5}}{2} x+\frac{1}{2}=-\frac{\sqrt{5}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{5}-1}{2} x=\frac{-\sqrt{5}-1}{2}$$