Solve 12/2m-2-m=3

$$\frac { 12 } { 2 m - 2 } - m = 3$$

Solve for m (complex solution)

$$m=\sqrt{10}-1\approx 2.16227766$$

$$m=-\left(\sqrt{10}+1\right)\approx -4.16227766$$

Steps Using the Quadratic Formula

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

$$12-m\times 2\left(m-1\right)=6\left(m-1\right)$$

Multiply -1 and 2 to get -2.

$$12-2m\left(m-1\right)=6\left(m-1\right)$$

Use the distributive property to multiply -2m by m-1.

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

Use the distributive property to multiply 6 by m-1.

$$12-2m^{2}+2m=6m-6$$

Subtract 6m from both sides.

$$12-2m^{2}+2m-6m=-6$$

Combine 2m and -6m to get -4m.

$$12-2m^{2}-4m=-6$$

Add 6 to both sides.

$$12-2m^{2}-4m+6=0$$

Add 12 and 6 to get 18.

$$18-2m^{2}-4m=0$$

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.

$$-2m^{2}-4m+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -4 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Square -4.

$$m=\frac{-\left(-4\right)±\sqrt{16-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$m=\frac{-\left(-4\right)±\sqrt{16+8\times 18}}{2\left(-2\right)}$$

Multiply 8 times 18.

$$m=\frac{-\left(-4\right)±\sqrt{16+144}}{2\left(-2\right)}$$

Add 16 to 144.

$$m=\frac{-\left(-4\right)±\sqrt{160}}{2\left(-2\right)}$$

Take the square root of 160.

$$m=\frac{-\left(-4\right)±4\sqrt{10}}{2\left(-2\right)}$$

The opposite of -4 is 4.

$$m=\frac{4±4\sqrt{10}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$m=\frac{4±4\sqrt{10}}{-4}$$

Now solve the equation m=\frac{4±4\sqrt{10}}{-4} when ± is plus. Add 4 to 4\sqrt{10}.

$$m=\frac{4\sqrt{10}+4}{-4}$$

Divide 4+4\sqrt{10} by -4.

$$m=-\left(\sqrt{10}+1\right)$$

Now solve the equation m=\frac{4±4\sqrt{10}}{-4} when ± is minus. Subtract 4\sqrt{10} from 4.

$$m=\frac{4-4\sqrt{10}}{-4}$$

Divide 4-4\sqrt{10} by -4.

$$m=\sqrt{10}-1$$

The equation is now solved.

$$m=-\left(\sqrt{10}+1\right) m=\sqrt{10}-1$$

Steps for Completing the Square

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

$$12-m\times 2\left(m-1\right)=6\left(m-1\right)$$

Multiply -1 and 2 to get -2.

$$12-2m\left(m-1\right)=6\left(m-1\right)$$

Use the distributive property to multiply -2m by m-1.

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

Use the distributive property to multiply 6 by m-1.

$$12-2m^{2}+2m=6m-6$$

Subtract 6m from both sides.

$$12-2m^{2}+2m-6m=-6$$

Combine 2m and -6m to get -4m.

$$12-2m^{2}-4m=-6$$

Subtract 12 from both sides.

$$-2m^{2}-4m=-6-12$$

Subtract 12 from -6 to get -18.

$$-2m^{2}-4m=-18$$

Divide both sides by -2.

$$\frac{-2m^{2}-4m}{-2}=-\frac{18}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$m^{2}+\left(-\frac{4}{-2}\right)m=-\frac{18}{-2}$$

Divide -4 by -2.

$$m^{2}+2m=-\frac{18}{-2}$$

Divide -18 by -2.

$$m^{2}+2m=9$$

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.

$$m^{2}+2m+1^{2}=9+1^{2}$$

Square 1.

$$m^{2}+2m+1=9+1$$

Add 9 to 1.

$$m^{2}+2m+1=10$$

Factor m^{2}+2m+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}.

$$\left(m+1\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+1\right)^{2}}=\sqrt{10}$$

Simplify.

$$m+1=\sqrt{10} m+1=-\sqrt{10}$$

Subtract 1 from both sides of the equation.

$$m=\sqrt{10}-1 m=-\sqrt{10}-1$$

Solve for m

$$m=\sqrt{10}-1\approx 2.16227766$$

$$m=-\sqrt{10}-1\approx -4.16227766$$

Steps Using the Quadratic Formula

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

$$12-m\times 2\left(m-1\right)=6\left(m-1\right)$$

Multiply -1 and 2 to get -2.

$$12-2m\left(m-1\right)=6\left(m-1\right)$$

Use the distributive property to multiply -2m by m-1.

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

Use the distributive property to multiply 6 by m-1.

$$12-2m^{2}+2m=6m-6$$

Subtract 6m from both sides.

$$12-2m^{2}+2m-6m=-6$$

Combine 2m and -6m to get -4m.

$$12-2m^{2}-4m=-6$$

Add 6 to both sides.

$$12-2m^{2}-4m+6=0$$

Add 12 and 6 to get 18.

$$18-2m^{2}-4m=0$$

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.

$$-2m^{2}-4m+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -4 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Square -4.

$$m=\frac{-\left(-4\right)±\sqrt{16-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$m=\frac{-\left(-4\right)±\sqrt{16+8\times 18}}{2\left(-2\right)}$$

Multiply 8 times 18.

$$m=\frac{-\left(-4\right)±\sqrt{16+144}}{2\left(-2\right)}$$

Add 16 to 144.

$$m=\frac{-\left(-4\right)±\sqrt{160}}{2\left(-2\right)}$$

Take the square root of 160.

$$m=\frac{-\left(-4\right)±4\sqrt{10}}{2\left(-2\right)}$$

The opposite of -4 is 4.

$$m=\frac{4±4\sqrt{10}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$m=\frac{4±4\sqrt{10}}{-4}$$

Now solve the equation m=\frac{4±4\sqrt{10}}{-4} when ± is plus. Add 4 to 4\sqrt{10}.

$$m=\frac{4\sqrt{10}+4}{-4}$$

Divide 4+4\sqrt{10} by -4.

$$m=-\left(\sqrt{10}+1\right)$$

Now solve the equation m=\frac{4±4\sqrt{10}}{-4} when ± is minus. Subtract 4\sqrt{10} from 4.

$$m=\frac{4-4\sqrt{10}}{-4}$$

Divide 4-4\sqrt{10} by -4.

$$m=\sqrt{10}-1$$

The equation is now solved.

$$m=-\left(\sqrt{10}+1\right) m=\sqrt{10}-1$$

Steps for Completing the Square

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

$$12-m\times 2\left(m-1\right)=6\left(m-1\right)$$

Multiply -1 and 2 to get -2.

$$12-2m\left(m-1\right)=6\left(m-1\right)$$

Use the distributive property to multiply -2m by m-1.

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

Use the distributive property to multiply 6 by m-1.

$$12-2m^{2}+2m=6m-6$$

Subtract 6m from both sides.

$$12-2m^{2}+2m-6m=-6$$

Combine 2m and -6m to get -4m.

$$12-2m^{2}-4m=-6$$

Subtract 12 from both sides.

$$-2m^{2}-4m=-6-12$$

Subtract 12 from -6 to get -18.

$$-2m^{2}-4m=-18$$

Divide both sides by -2.

$$\frac{-2m^{2}-4m}{-2}=-\frac{18}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$m^{2}+\left(-\frac{4}{-2}\right)m=-\frac{18}{-2}$$

Divide -4 by -2.

$$m^{2}+2m=-\frac{18}{-2}$$

Divide -18 by -2.

$$m^{2}+2m=9$$

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.

$$m^{2}+2m+1^{2}=9+1^{2}$$

Square 1.

$$m^{2}+2m+1=9+1$$

Add 9 to 1.

$$m^{2}+2m+1=10$$

Factor m^{2}+2m+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}.

$$\left(m+1\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+1\right)^{2}}=\sqrt{10}$$

Simplify.

$$m+1=\sqrt{10} m+1=-\sqrt{10}$$

Subtract 1 from both sides of the equation.

$$m=\sqrt{10}-1 m=-\sqrt{10}-1$$