Math Problem Statement

y to the second power minus six plus eight over three y plus 6 times y plus two over three y minus twelve

Solution

You provided the expression:

y26+83y+6×y+23y12y^2 - 6 + \frac{8}{3y} + 6 \times \frac{y+2}{3y} - 12

Let's simplify it step by step.

Step 1: Combine Like Terms

The constant terms are:
6+612=12-6 + 6 - 12 = -12

So the expression becomes:

y212+83y+6×y+23yy^2 - 12 + \frac{8}{3y} + 6 \times \frac{y+2}{3y}

Step 2: Distribute the 6

6×y+23y=6(y+2)3y=6y+123y=2y+4y6 \times \frac{y+2}{3y} = \frac{6(y+2)}{3y} = \frac{6y + 12}{3y} = \frac{2y + 4}{y}

Step 3: Rewrite the Expression

y212+83y+2y+4yy^2 - 12 + \frac{8}{3y} + \frac{2y + 4}{y}

Step 4: Express Everything with a Common Denominator

The least common denominator (LCD) is 3y3y, so we rewrite terms:

  • 2y+4y=3(2y+4)3y=6y+123y\frac{2y+4}{y} = \frac{3(2y+4)}{3y} = \frac{6y + 12}{3y}
  • 83y\frac{8}{3y} stays the same.

Thus, the expression is:

y212+6y+12+83yy^2 - 12 + \frac{6y + 12 + 8}{3y}

y212+6y+203yy^2 - 12 + \frac{6y + 20}{3y}

Final Answer:

y212+6y+203yy^2 - 12 + \frac{6y + 20}{3y}

Would you like me to further simplify or factor any part?

Related Questions:

  1. How do you find the least common denominator for rational expressions?
  2. How can this expression be rewritten in terms of partial fractions?
  3. What happens if we set this expression equal to zero and solve for yy?
  4. How do you simplify algebraic fractions involving polynomials?
  5. What are the restrictions on yy for this expression to be valid?

Tip:

When working with fractions in algebraic expressions, always look for the least common denominator to simplify operations efficiently!

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Simplification of Algebraic Expressions

Formulas

Distributive property
Least common denominator (LCD)

Theorems

Simplifying rational expressions
Combining like terms

Suitable Grade Level

Grades 9-11