Solve Matrix

$$\left. \begin{array} { l } { \text { If } m + \frac { 1 } { m - 2 } = 4 \text { then } } \\ { ( m - 2 ) ^ { 2 } + \frac { 1 } { ( m - 2 ) ^ { 2 } } } \\ { \left. \begin{array} { l l } { \text { (A) } - 2 } & { \text { (B) } 0 } \\ { \text { (C) } 2 } & { \text { (D) } 4 } \end{array} \right. } \end{array} \right.$$

Solve for m

$$m=3$$

Steps Using the Quadratic Formula

Variable m cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by m-2.

$$\left(m-2\right)m+1=4\left(m-2\right)$$

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

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

Use the distributive property to multiply 4 by m-2.

$$m^{2}-2m+1=4m-8$$

Subtract 4m from both sides.

$$m^{2}-2m+1-4m=-8$$

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

$$m^{2}-6m+1=-8$$

Add 8 to both sides.

$$m^{2}-6m+1+8=0$$

Add 1 and 8 to get 9.

$$m^{2}-6m+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9}}{2}$$

Square -6.

$$m=\frac{-\left(-6\right)±\sqrt{36-4\times 9}}{2}$$

Multiply -4 times 9.

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

Add 36 to -36.

$$m=\frac{-\left(-6\right)±\sqrt{0}}{2}$$

Take the square root of 0.

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

The opposite of -6 is 6.

$$m=\frac{6}{2}$$

Divide 6 by 2.

$$m=3$$

Steps for Completing the Square

Variable m cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by m-2.

$$\left(m-2\right)m+1=4\left(m-2\right)$$

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

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

Use the distributive property to multiply 4 by m-2.

$$m^{2}-2m+1=4m-8$$

Subtract 4m from both sides.

$$m^{2}-2m+1-4m=-8$$

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

$$m^{2}-6m+1=-8$$

Subtract 1 from both sides.

$$m^{2}-6m=-8-1$$

Subtract 1 from -8 to get -9.

$$m^{2}-6m=-9$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-6m+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Square -3.

$$m^{2}-6m+9=-9+9$$

Add -9 to 9.

$$m^{2}-6m+9=0$$

Factor m^{2}-6m+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m-3=0 m-3=0$$

Add 3 to both sides of the equation.

$$m=3 m=3$$

The equation is now solved. Solutions are the same.

$$m=3$$