Math Problem Statement

Factor the deference of two square

Solution

To factor the difference of two squares, we use the formula:

a2b2=(a+b)(ab)a^2 - b^2 = (a + b)(a - b)

This formula states that any expression of the form a2b2a^2 - b^2 can be factored into the product of two binomials: one with a sum and the other with a difference.

Example:

Consider the expression x29x^2 - 9.

  1. Identify aa and bb:

    • Here, x2x^2 is a2a^2, so a=xa = x.
    • 99 is b2b^2, so b=3b = 3.
  2. Apply the formula: x29=(x+3)(x3)x^2 - 9 = (x + 3)(x - 3)

So, the factorization of x29x^2 - 9 is (x+3)(x3)(x + 3)(x - 3).

Another Example:

Factor the expression 16y225z216y^2 - 25z^2.

  1. Identify aa and bb:

    • 16y216y^2 is (4y)2(4y)^2, so a=4ya = 4y.
    • 25z225z^2 is (5z)2(5z)^2, so b=5zb = 5z.
  2. Apply the formula: 16y225z2=(4y+5z)(4y5z)16y^2 - 25z^2 = (4y + 5z)(4y - 5z)

So, the factorization of 16y225z216y^2 - 25z^2 is (4y+5z)(4y5z)(4y + 5z)(4y - 5z).

Do you want details or have any questions? Here are five questions for further exploration:

  1. How can you recognize the difference of two squares in an algebraic expression?
  2. Can the difference of two squares be used in simplifying complex algebraic fractions?
  3. What happens if you attempt to factor the sum of two squares?
  4. How is the difference of squares used in solving quadratic equations?
  5. Can the difference of two squares formula be extended to higher powers?

Tip: Always check if an expression can be written in the form of a2b2a^2 - b^2 before applying the difference of squares formula.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factorization
Quadratic Equations

Formulas

Difference of squares formula (a^2 - b^2 = (a + b)(a - b))

Theorems

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Suitable Grade Level

Grades 9-12