Question

evaluate
independent practice
learning goal
i can solve rational equations in one variable and determine extraneous solutions. i can solve radical equations in one variable and determine extraneous solutions. i can explain how extraneous solutions may arise from rational or radical equations.
how i did (circle one)
starting... getting there... got it!
lesson 9.3 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.
solve each rational equation algebraically.

1. $\frac{-x + 6}{x^{2}-3x + 2}=\frac{x + 6}{3x-3}$

$x
eq1,2$
what is a possible first step to solving the rational expression?

1. $\frac{3}{x + 2}+\frac{3}{2x + 4}=\frac{x}{2x + 4}$

find the common denominators.
cross multiply.
use the quadratic formula.
add the numerator and denominators.

Explanation:

Step1: Factor denominators

Factor $x^{2}-3x + 2=(x - 1)(x - 2)$ and $3x-3 = 3(x - 1)$. The equation $\frac{-x + 6}{x^{2}-3x + 2}=\frac{x + 6}{3x-3}$ becomes $\frac{-x + 6}{(x - 1)(x - 2)}=\frac{x + 6}{3(x - 1)}$.

Step2: Cross - multiply

Cross - multiplying gives $3(-x + 6)(x - 1)=(x + 6)(x - 1)(x - 2)$. Since $x
eq1$, we can divide both sides by $(x - 1)$ (as long as $x
eq1$ which is in the domain restriction). We get $3(-x + 6)=(x + 6)(x - 2)$.

Step3: Expand both sides

Expand: $-3x+18=x^{2}-2x+6x - 12$.

Step4: Rearrange to quadratic form

Rearrange to $x^{2}+5x - 30 = 0$.

Step5: Solve quadratic equation

Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b = 5$, $c=-30$.
$x=\frac{-5\pm\sqrt{5^{2}-4\times1\times(-30)}}{2\times1}=\frac{-5\pm\sqrt{25 + 120}}{2}=\frac{-5\pm\sqrt{145}}{2}$.

For the second question:

The denominators of the rational expression $\frac{3}{x + 2}+\frac{3}{2x+4}=\frac{x}{2x + 4}$ are $x + 2$ and $2x+4=2(x + 2)$. The first step is to find the common denominators. Since the common denominator of the left - hand side is $2(x + 2)$, we can rewrite the left - hand side expressions with the common denominator.

Answer:

1. $x=\frac{-5\pm\sqrt{145}}{2}$
2. Find the common denominators.