Solve x-2/4=9/x-2

$$\frac{ x-2 }{ 4 } = \frac{ 9 }{ x-2 }$$

Solve for x

$$x=-4$$

$$x=8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 4\left(x-2\right), the least common multiple of 4,x-2.

$$\left(x-2\right)\left(x-2\right)=4\times 9$$

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

$$\left(x-2\right)^{2}=4\times 9$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4=4\times 9$$

Multiply 4 and 9 to get 36.

$$x^{2}-4x+4=36$$

Subtract 36 from both sides.

$$x^{2}-4x+4-36=0$$

Subtract 36 from 4 to get -32.

$$x^{2}-4x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-32\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$x=\frac{-\left(-4\right)±\sqrt{16+128}}{2}$$

Add 16 to 128.

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

Take the square root of 144.

$$x=\frac{-\left(-4\right)±12}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±12}{2}$$

Now solve the equation x=\frac{4±12}{2} when ± is plus. Add 4 to 12.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{4±12}{2} when ± is minus. Subtract 12 from 4.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=8 x=-4$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 4\left(x-2\right), the least common multiple of 4,x-2.

$$\left(x-2\right)\left(x-2\right)=4\times 9$$

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

$$\left(x-2\right)^{2}=4\times 9$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4=4\times 9$$

Multiply 4 and 9 to get 36.

$$x^{2}-4x+4=36$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-2=6 x-2=-6$$

Add 2 to both sides of the equation.

$$x=8 x=-4$$