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

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

Solve for x

$$x=\frac{\sqrt{42}}{6}+1\approx 2.08012345$$

$$x=-\frac{\sqrt{42}}{6}+1\approx -0.08012345$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply 2 by 3x-5.

$$\left(6x-10\right)x=2x+1$$

Use the distributive property to multiply 6x-10 by x.

$$6x^{2}-10x=2x+1$$

Subtract 2x from both sides.

$$6x^{2}-10x-2x=1$$

Combine -10x and -2x to get -12x.

$$6x^{2}-12x=1$$

Subtract 1 from both sides.

$$6x^{2}-12x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -12 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-12\right)±\sqrt{144-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+24}}{2\times 6}$$

Add 144 to 24.

$$x=\frac{-\left(-12\right)±\sqrt{168}}{2\times 6}$$

Take the square root of 168.

$$x=\frac{-\left(-12\right)±2\sqrt{42}}{2\times 6}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{42}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{12±2\sqrt{42}}{12}$$

Now solve the equation x=\frac{12±2\sqrt{42}}{12} when ± is plus. Add 12 to 2\sqrt{42}.

$$x=\frac{2\sqrt{42}+12}{12}$$

Divide 12+2\sqrt{42} by 12.

$$x=\frac{\sqrt{42}}{6}+1$$

Now solve the equation x=\frac{12±2\sqrt{42}}{12} when ± is minus. Subtract 2\sqrt{42} from 12.

$$x=\frac{12-2\sqrt{42}}{12}$$

Divide 12-2\sqrt{42} by 12.

$$x=-\frac{\sqrt{42}}{6}+1$$

The equation is now solved.

$$x=\frac{\sqrt{42}}{6}+1 x=-\frac{\sqrt{42}}{6}+1$$

Steps for Completing the Square

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

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

Use the distributive property to multiply 2 by 3x-5.

$$\left(6x-10\right)x=2x+1$$

Use the distributive property to multiply 6x-10 by x.

$$6x^{2}-10x=2x+1$$

Subtract 2x from both sides.

$$6x^{2}-10x-2x=1$$

Combine -10x and -2x to get -12x.

$$6x^{2}-12x=1$$

Divide both sides by 6.

$$\frac{6x^{2}-12x}{6}=\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{12}{6}\right)x=\frac{1}{6}$$

Divide -12 by 6.

$$x^{2}-2x=\frac{1}{6}$$

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

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

Add \frac{1}{6} to 1.

$$x^{2}-2x+1=\frac{7}{6}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=\frac{\sqrt{42}}{6} x-1=-\frac{\sqrt{42}}{6}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{42}}{6}+1 x=-\frac{\sqrt{42}}{6}+1$$