Simplify (u^2-9u+18)/(45-5u^2)

The problem presented is an algebraic simplification task. It requires you to simplify a rational expression, which means reducing the expression to its simplest form. This is done by identifying and canceling out any common factors in the numerator (u^2-9u+18) and the denominator (45-5u^2). It may involve factoring polynomials and finding common factors that can be eliminated. The goal is to rewrite the expression such that no further simplification is possible.

$\frac{u^{2} - 9 u + 18}{45 - 5 u^{2}}$

Answer

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Solution:

Step 1: Factor the numerator.

Step 1.1: Apply the AC method.

Identify two numbers whose product equals $18$ and sum equals $-9$. These numbers are $-6$ and $-3$.

Step 1.2: Express the factored form.

The factored form of the numerator is $\frac{(u - 6)(u - 3)}{45 - 5u^2}$.

Step 2: Factor the denominator.

Step 2.1: Factor out the common factor in the denominator.

Step 2.1.1: Extract $5$ from $45$.

The expression becomes $\frac{(u - 6)(u - 3)}{5(9) - 5u^2}$.

Step 2.1.2: Extract $5$ from $-5u^2$.

The expression is now $\frac{(u - 6)(u - 3)}{5(9) + 5(-u^2)}$.

Step 2.1.3: Factor out $5$ completely.

The denominator is factored to $5(9 - u^2)$.

Step 2.2: Recognize the difference of squares.

Rewrite $9$ as $3^2$ to get $\frac{(u - 6)(u - 3)}{5(3^2 - u^2)}$.

Step 2.3: Factor using the difference of squares formula.

The denominator is factored to $5(3 + u)(3 - u)$.

Step 3: Simplify the expression by canceling common factors.

Step 3.1: Cancel out $(u - 3)$ and $(3 - u)$.

Step 3.1.1: Factor $-1$ from $u$.

The expression becomes $\frac{(u - 6)(-1)(-u - 3)}{5(3 + u)(3 - u)}$.

Step 3.1.2: Rewrite $-3$ as $-1(3)$.

This gives us $\frac{(u - 6)(-1)(-u - 1(3))}{5(3 + u)(3 - u)}$.

Step 3.1.3: Factor $-1$ out completely.

The numerator simplifies to $\frac{(u - 6)(-1)(-u + 3)}{5(3 + u)(3 - u)}$.

Step 3.1.4: Reorder terms.

The expression is reordered to $\frac{(u - 6)(-1)(3 - u)}{5(3 + u)(3 - u)}$.

Step 3.1.5: Cancel the common factor.

After canceling, we get $\frac{(u - 6)(-1)}{5(3 + u)}$.

Step 3.1.6: Rewrite the simplified expression.

The final simplified form is $\frac{(-1)(u - 6)}{5(3 + u)}$.

Step 3.2: Final simplification.

Step 3.2.1: Move $-1$ to the front.

The expression is now $\frac{-1(u - 6)}{5(3 + u)}$.

Step 3.2.2: Simplify by moving the negative sign in front.

The final result is $-\frac{u - 6}{5(3 + u)}$.

Knowledge Notes:

To simplify a rational expression, follow these steps:

Factorization: Break down both the numerator and the denominator into their simplest factors. For quadratic expressions, methods like factoring by grouping, the AC method, or the difference of squares can be used.
Common Factors: Look for and cancel out any common factors in the numerator and the denominator. This step is crucial for simplification.
Difference of Squares: Recognize when the expression involves a difference of squares, which is an expression of the form $a^2 - b^2$. This can be factored into $(a + b)(a - b)$.
Negative Signs: Be careful with negative signs. Factoring out $-1$ can help to recognize common factors that might not be immediately obvious.
Simplification: After canceling common factors, rewrite the expression in its simplest form. If there's a negative sign in the numerator or denominator, it can be moved in front of the fraction.
Latex Formatting: When writing mathematical expressions, especially for complex fractions and factoring, use Latex to clearly convey the steps and the structure of the expression.