Math Problem Statement

Écrire l'expression suivante sous la forme d'une seule fraction avec le numérateur et le dénominateur factorisés et simplifiés au maximum: 12.5/x - 10 + 4x/2 pour x ≠ 0

Solution

The expression in the image asks to simplify the following equation into a single fraction, with both the numerator and the denominator factored and simplified:

12.5x10+4x2,for x0\frac{12.5}{x} - 10 + \frac{4x}{2}, \quad \text{for } x \neq 0

Step 1: Simplify the terms

  • The second term 1010 is already simplified.
  • The third term 4x2\frac{4x}{2} simplifies to 2x2x.

Thus, the expression becomes: 12.5x10+2x\frac{12.5}{x} - 10 + 2x

Step 2: Express all terms with a common denominator

To combine all terms into a single fraction, the common denominator is xx. The terms must be rewritten as:

  • 12.5x\frac{12.5}{x} remains the same.
  • Rewrite 1010 as 10xx\frac{10x}{x}.
  • Rewrite 2x2x as 2x2x\frac{2x^2}{x}.

Thus, the expression becomes: 12.5x10xx+2x2x\frac{12.5}{x} - \frac{10x}{x} + \frac{2x^2}{x}

Step 3: Combine the terms

Now that the terms share the same denominator, combine the numerators: 12.510x+2x2x\frac{12.5 - 10x + 2x^2}{x}

Final Expression:

The simplified expression in a single fraction is: 2x210x+12.5x\frac{2x^2 - 10x + 12.5}{x}

If you'd like further details or have any questions, feel free to ask!

Related Questions:

  1. How do you factor quadratic expressions like 2x210x+12.52x^2 - 10x + 12.5?
  2. What steps should be taken if the denominator is a binomial instead of a single variable?
  3. How would you simplify expressions with different denominators?
  4. Can we apply this method to complex numbers or polynomials with higher degrees?
  5. How does dividing by zero affect the expression?

Tip:

When simplifying expressions, always look for a common denominator early to reduce the number of operations required later.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Simplification of algebraic fractions
Combining like terms
Factoring

Formulas

a/b + c/b = (a + c)/b
Quadratic expression: ax^2 + bx + c

Theorems

Distributive property of multiplication
Quadratic formula: (-b ± √(b²-4ac)) / 2a

Suitable Grade Level

Grades 10-12