Solve 1+3x/left(x-1right)=x/2

$$\frac{ 1+3x }{ \left( x-1 \right) } = \frac{ x }{ 2 }$$

Solve for x

$$x = \frac{\sqrt{57} + 7}{2} \approx 7.274917218$$

$$x=\frac{7-\sqrt{57}}{2}\approx -0.274917218$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right), the least common multiple of x-1,2.

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

Use the distributive property to multiply 2 by 1+3x.

$$2+6x=\left(x-1\right)x$$

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

Combine 6x and x to get 7x.

$$2+7x-x^{2}=0$$

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.

$$-x^{2}+7x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-7±\sqrt{49+4\times 2}}{2\left(-1\right)}$$

Multiply 4 times 2.

$$x=\frac{-7±\sqrt{49+8}}{2\left(-1\right)}$$

Add 49 to 8.

$$x=\frac{-7±\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-7±\sqrt{57}}{-2}$$

Now solve the equation x=\frac{-7±\sqrt{57}}{-2} when ± is plus. Add -7 to \sqrt{57}.

$$x=\frac{\sqrt{57}-7}{-2}$$

Divide -7+\sqrt{57} by -2.

$$x=\frac{7-\sqrt{57}}{2}$$

Now solve the equation x=\frac{-7±\sqrt{57}}{-2} when ± is minus. Subtract \sqrt{57} from -7.

$$x=\frac{-\sqrt{57}-7}{-2}$$

Divide -7-\sqrt{57} by -2.

$$x=\frac{\sqrt{57}+7}{2}$$

The equation is now solved.

$$x=\frac{7-\sqrt{57}}{2} x=\frac{\sqrt{57}+7}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right), the least common multiple of x-1,2.

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

Use the distributive property to multiply 2 by 1+3x.

$$2+6x=\left(x-1\right)x$$

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

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

Subtract x^{2} from both sides.

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

Add x to both sides.

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

Combine 6x and x to get 7x.

$$2+7x-x^{2}=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$7x-x^{2}=-2$$

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+7x}{-1}=-\frac{2}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{7}{-1}x=-\frac{2}{-1}$$

Divide 7 by -1.

$$x^{2}-7x=-\frac{2}{-1}$$

Divide -2 by -1.

$$x^{2}-7x=2$$

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

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

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=2+\frac{49}{4}$$

Add 2 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{57}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{57}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{57}}{2} x-\frac{7}{2}=-\frac{\sqrt{57}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{\sqrt{57}+7}{2} x=\frac{7-\sqrt{57}}{2}$$