Math Problem Statement

solve by factoring: x2 - 5x + 6 = 0

Solution

To solve the quadratic equation x25x+6=0x^2 - 5x + 6 = 0 by factoring, we need to find two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of the linear term, xx).

Step-by-step solution:

  1. Identify the factors of the constant term (6): The factors of 6 are: (1,6),(2,3)(1, 6), (2, 3)

  2. Find the pair that adds to -5: The pair 2-2 and 3-3 multiply to give 6 and add up to -5.

  3. Factor the quadratic: We can rewrite the equation as: x22x3x+6=0x^2 - 2x - 3x + 6 = 0 Now, group the terms: (x22x)(3x6)=0(x^2 - 2x) - (3x - 6) = 0 Factor each group: x(x2)3(x2)=0x(x - 2) - 3(x - 2) = 0 Factor out the common binomial (x2)(x - 2): (x2)(x3)=0(x - 2)(x - 3) = 0

  4. Solve for xx: Set each factor equal to zero: x2=0orx3=0x - 2 = 0 \quad \text{or} \quad x - 3 = 0 Solving these gives: x=2orx=3x = 2 \quad \text{or} \quad x = 3

Final solution:

The solutions are: x=2orx=3x = 2 \quad \text{or} \quad x = 3

Would you like a more detailed explanation of any of the steps? Here are some related questions for you:

  1. What would the graph of x25x+6=0x^2 - 5x + 6 = 0 look like?
  2. How would you solve a quadratic equation using the quadratic formula instead of factoring?
  3. What happens if the equation does not factor easily?
  4. How do you identify when a quadratic equation is prime (cannot be factored)?
  5. What are the conditions for factoring quadratics with negative coefficients?

Tip: When factoring quadratics, always start by looking for a pair of numbers that multiply to the constant term and add to the coefficient of the linear term.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Factoring

Formulas

Quadratic equation ax^2 + bx + c = 0
Factoring method

Theorems

Zero Product Property

Suitable Grade Level

Grades 7-9