Math Problem Statement

2x^2+7x+3/4x+12 perform all operations and simplify

Solution

To simplify the expression 2x2+7x+34x+12\frac{2x^2 + 7x + 3}{4x + 12}, we will follow these steps:

  1. Factorize both the numerator and the denominator, if possible.
  2. Simplify by canceling common factors.

Let's start by factorizing the denominator.

Step 1: Factorize the Denominator

The denominator is 4x+124x + 12. We can factor out the common factor of 4: 4x+12=4(x+3)4x + 12 = 4(x + 3)

Step 2: Factorize the Numerator

The numerator is 2x2+7x+32x^2 + 7x + 3. We need to factorize this quadratic expression.

To factorize 2x2+7x+32x^2 + 7x + 3:

  • Look for two numbers that multiply to 23=62 \cdot 3 = 6 and add up to 7.
  • These numbers are 6 and 1.

Thus, we rewrite the middle term 7x7x as 6x+x6x + x: 2x2+7x+3=2x2+6x+x+32x^2 + 7x + 3 = 2x^2 + 6x + x + 3

Now, group the terms: 2x2+6x+x+3=2x(x+3)+1(x+3)2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3)

Factor out the common binomial factor (x+3)(x + 3): 2x(x+3)+1(x+3)=(2x+1)(x+3)2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)

Step 3: Simplify the Expression

Now the expression becomes: 2x2+7x+34x+12=(2x+1)(x+3)4(x+3)\frac{2x^2 + 7x + 3}{4x + 12} = \frac{(2x + 1)(x + 3)}{4(x + 3)}

Cancel the common factor (x+3)(x + 3): (2x+1)(x+3)4(x+3)=2x+14\frac{(2x + 1)(x + 3)}{4(x + 3)} = \frac{2x + 1}{4}

So, the simplified expression is: 2x+14\frac{2x + 1}{4}

Would you like more details or have any questions?

Here are some questions you might want to ask:

  1. How do you factorize a quadratic expression?
  2. What if the numerator or denominator cannot be factorized?
  3. Can you show more examples of rational expressions?
  4. How do you handle complex rational expressions?
  5. What is the difference between simplifying and solving an expression?

Tip: Always check if the terms in the numerator and denominator can be factorized before attempting to simplify rational expressions. This can help you identify and cancel common factors easily.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring

Formulas

Quadratic formula

Theorems

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Suitable Grade Level

Grades 9-12