Algebra for prep 2

🧮 Math Problem: Difference of Squares

Subject: Algebra | Grade: 9th Grade | Semester: First Term | Curriculum: Egyptian Curriculum

Problem Statement

If \(x = (\sqrt{5} - 2)\) and \(y = (\sqrt{5} + 2)\), find the value of the expression \(x^2 - y^2\) in its simplest form.

Final Answer

The value of the expression is \(-8\sqrt{5}\).

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Step-by-Step Solution

Method 1: Using the Difference of Squares

A quick way to solve this is to use the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).

• 1. Calculate (x - y): \((\sqrt{5} - 2) - (\sqrt{5} + 2) = -4\)
• 2. Calculate (x + y): \((\sqrt{5} - 2) + (\sqrt{5} + 2) = 2\sqrt{5}\)
• 3. Multiply results: \((x-y)(x+y) = (-4)(2\sqrt{5}) = -8\sqrt{5}\)

Method 2: Direct Calculation

You can also solve this by squaring each term first and then subtracting.

• 1. Calculate \(x^2\):

  \((\sqrt{5} - 2)^2 = (\sqrt{5})^2 - 2(\sqrt{5})(2) + (2)^2\)

  \( = 5 - 4\sqrt{5} + 4\)

  \( = 9 - 4\sqrt{5}\)

• 2. Calculate \(y^2\):

  \((\sqrt{5} + 2)^2 = (\sqrt{5})^2 + 2(\sqrt{5})(2) + (2)^2\)

  \( = 5 + 4\sqrt{5} + 4\)

  \( = 9 + 4\sqrt{5}\)

• 3. Subtract \(y^2\) from \(x^2\):

  \((9 - 4\sqrt{5}) - (9 + 4\sqrt{5})\)

  \( = 9 - 4\sqrt{5} - 9 - 4\sqrt{5}\)

  \( = -8\sqrt{5}\)