Solve m+2/m-3=-m+1/m-2

$$\frac { m + 2 } { m - 3 } = \frac { - m + 1 } { m - 2 }$$

Solve for m

$$m=\frac{\sqrt{6}}{2}+1\approx 2.224744871$$

$$m=-\frac{\sqrt{6}}{2}+1\approx -0.224744871$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by \left(m-3\right)\left(m-2\right), the least common multiple of m-3,m-2.

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

Consider \left(m-2\right)\left(m+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$m^{2}-4=\left(m-3\right)\left(-m+1\right)$$

Use the distributive property to multiply m-3 by -m+1.

$$m^{2}-4=m\left(-m\right)+m-3\left(-m\right)-3$$

Multiply -3 and -1 to get 3.

$$m^{2}-4=m\left(-m\right)+m+3m-3$$

Combine m and 3m to get 4m.

$$m^{2}-4=m\left(-m\right)+4m-3$$

Subtract m\left(-m\right) from both sides.

$$m^{2}-4-m\left(-m\right)=4m-3$$

Subtract 4m from both sides.

$$m^{2}-4-m\left(-m\right)-4m=-3$$

Add 3 to both sides.

$$m^{2}-4-m\left(-m\right)-4m+3=0$$

Multiply m and m to get m^{2}.

$$m^{2}-4-m^{2}\left(-1\right)-4m+3=0$$

Multiply -1 and -1 to get 1.

$$m^{2}-4+m^{2}-4m+3=0$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$2m^{2}-4-4m+3=0$$

Add -4 and 3 to get -1.

$$2m^{2}-1-4m=0$$

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.

$$2m^{2}-4m-1=0$$

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

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

Square -4.

$$m=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-1\right)}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-\left(-4\right)±\sqrt{16-8\left(-1\right)}}{2\times 2}$$

Multiply -8 times -1.

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

Add 16 to 8.

$$m=\frac{-\left(-4\right)±\sqrt{24}}{2\times 2}$$

Take the square root of 24.

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

The opposite of -4 is 4.

$$m=\frac{4±2\sqrt{6}}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{4±2\sqrt{6}}{4}$$

Now solve the equation m=\frac{4±2\sqrt{6}}{4} when ± is plus. Add 4 to 2\sqrt{6}.

$$m=\frac{2\sqrt{6}+4}{4}$$

Divide 4+2\sqrt{6} by 4.

$$m=\frac{\sqrt{6}}{2}+1$$

Now solve the equation m=\frac{4±2\sqrt{6}}{4} when ± is minus. Subtract 2\sqrt{6} from 4.

$$m=\frac{4-2\sqrt{6}}{4}$$

Divide 4-2\sqrt{6} by 4.

$$m=-\frac{\sqrt{6}}{2}+1$$

The equation is now solved.

$$m=\frac{\sqrt{6}}{2}+1 m=-\frac{\sqrt{6}}{2}+1$$

Steps for Completing the Square

Variable m cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by \left(m-3\right)\left(m-2\right), the least common multiple of m-3,m-2.

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

Consider \left(m-2\right)\left(m+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$m^{2}-4=\left(m-3\right)\left(-m+1\right)$$

Use the distributive property to multiply m-3 by -m+1.

$$m^{2}-4=m\left(-m\right)+m-3\left(-m\right)-3$$

Multiply -3 and -1 to get 3.

$$m^{2}-4=m\left(-m\right)+m+3m-3$$

Combine m and 3m to get 4m.

$$m^{2}-4=m\left(-m\right)+4m-3$$

Subtract m\left(-m\right) from both sides.

$$m^{2}-4-m\left(-m\right)=4m-3$$

Subtract 4m from both sides.

$$m^{2}-4-m\left(-m\right)-4m=-3$$

Multiply m and m to get m^{2}.

$$m^{2}-4-m^{2}\left(-1\right)-4m=-3$$

Multiply -1 and -1 to get 1.

$$m^{2}-4+m^{2}-4m=-3$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$2m^{2}-4-4m=-3$$

Add 4 to both sides.

$$2m^{2}-4m=-3+4$$

Add -3 and 4 to get 1.

$$2m^{2}-4m=1$$

Divide both sides by 2.

$$\frac{2m^{2}-4m}{2}=\frac{1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}+\left(-\frac{4}{2}\right)m=\frac{1}{2}$$

Divide -4 by 2.

$$m^{2}-2m=\frac{1}{2}$$

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.

$$m^{2}-2m+1=\frac{1}{2}+1$$

Add \frac{1}{2} to 1.

$$m^{2}-2m+1=\frac{3}{2}$$

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}.

$$\left(m-1\right)^{2}=\frac{3}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-1\right)^{2}}=\sqrt{\frac{3}{2}}$$

Simplify.

$$m-1=\frac{\sqrt{6}}{2} m-1=-\frac{\sqrt{6}}{2}$$

Add 1 to both sides of the equation.

$$m=\frac{\sqrt{6}}{2}+1 m=-\frac{\sqrt{6}}{2}+1$$