Question

1. $\frac{12}{x}=\frac{x-2}{3}$

excluded values $x\
eq$ ________
answer: $x=$ ________

Explanation:

Step1: Find excluded value

Excluded values make denominator 0. For $\frac{12}{x}$, $x=0$ is invalid.

Step2: Cross-multiply to eliminate fractions

$$12 \times 3 = x(x-2)$$
$$36 = x^2 - 2x$$

Step3: Rearrange to quadratic equation

$$x^2 - 2x - 36 = 0$$

Step4: Solve quadratic via quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=1, b=-2, c=-36$:
$$x = \frac{2\pm\sqrt{(-2)^2-4(1)(-36)}}{2(1)}$$
$$x = \frac{2\pm\sqrt{4+144}}{2}$$
$$x = \frac{2\pm\sqrt{148}}{2}$$
$$x = \frac{2\pm2\sqrt{37}}{2}$$
$$x = 1\pm\sqrt{37}$$

Answer:

Excluded Values $x
eq 0$