Math Problem Statement

Add: (2 / (3x^2 - 10x + 3)) + (1 / (x - 3)). Simplify your answer as much as possible.

Solution

To solve the given expression, we need to add the two fractions:

23x210x+3+1x3\frac{2}{3x^2 - 10x + 3} + \frac{1}{x - 3}

Steps to simplify:

  1. Factor the denominator of the first fraction: 3x210x+33x^2 - 10x + 3.

    • Factor this quadratic expression using techniques like factoring by grouping or the quadratic formula.
  2. Find a common denominator between the two fractions. The common denominator will be the least common multiple (LCM) of the factored form of 3x210x+33x^2 - 10x + 3 and x3x - 3.

  3. Combine the fractions over the common denominator.

  4. Simplify the resulting expression if possible.

Step 1: Factor the quadratic

We factor 3x210x+33x^2 - 10x + 3:

  • We look for two numbers that multiply to 3×3=93 \times 3 = 9 and add to 10-10. These numbers are 9-9 and 1-1.
  • So we can factor as: 3x210x+3=(3x1)(x3)3x^2 - 10x + 3 = (3x - 1)(x - 3)

Step 2: Rewrite the expression

Now the expression becomes: 2(3x1)(x3)+1x3\frac{2}{(3x - 1)(x - 3)} + \frac{1}{x - 3}

Step 3: Find a common denominator

The common denominator is (3x1)(x3)(3x - 1)(x - 3). Rewrite the second fraction to have this denominator: 1x3=3x1(3x1)(x3)\frac{1}{x - 3} = \frac{3x - 1}{(3x - 1)(x - 3)}

Step 4: Add the fractions

Now that the denominators are the same, we can add the numerators: 2(3x1)(x3)+3x1(3x1)(x3)=2+(3x1)(3x1)(x3)\frac{2}{(3x - 1)(x - 3)} + \frac{3x - 1}{(3x - 1)(x - 3)} = \frac{2 + (3x - 1)}{(3x - 1)(x - 3)}

Simplify the numerator: 2+(3x1)=3x+12 + (3x - 1) = 3x + 1

So the result is: 3x+1(3x1)(x3)\frac{3x + 1}{(3x - 1)(x - 3)}

Final Answer:

3x+1(3x1)(x3)\frac{3x + 1}{(3x - 1)(x - 3)}

This is the simplified form of the sum of the two fractions.

Would you like any further details or clarifications on any of the steps?


Here are five related questions for further practice:

  1. How do you factor quadratics like 3x210x+33x^2 - 10x + 3 in general?
  2. What is the process for finding a common denominator in fraction addition?
  3. How can you simplify complex rational expressions with quadratic terms?
  4. What other methods can be used if factoring by grouping doesn't work?
  5. What strategies can help verify if the simplification is correct?

Tip: Always check if a quadratic can be factored first, as it simplifies further operations like finding common denominators!

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring Quadratics
Fraction Addition

Formulas

Factoring Quadratics: ax^2 + bx + c = 0
Fraction Addition: a/b + c/d = (ad + bc) / bd

Theorems

Factoring Theorem
Least Common Denominator

Suitable Grade Level

Grades 10-12