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Square of a Binomial

Learn how to use binomial formulas like (a + b)², (a − b)², and (a + b)(a − b) to simplify expressions. Let’s get started! 🚀

Square of a Binomial - introduction visual

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Flashcards

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Expansion and simplification of (a + b)² to a² + 2ab + b² using distributive method.Square of a binomial formula, showing (a + b)² = a² + b² + 2ab with coloured squares and rectangles representing each term.Three binomial square formulas, including (a + b)² = a² + b² + 2ab, (a - b)² = a² + b² - 2ab, and (a + b)(a - b) = a² - b².Binomial expansion showing the formula (a + b)² = a² + b² + 2ab, and an example (-2x + 3)² = 4x² + 9 - 12x.Expanding the binomial (5x - 2y)² using the formula (a - b)² = a² + b² - 2ab, resulting in 25x² + 4y² - 20xy.Identifying variables a and b with signs and coefficients in the formula (a + b)(a - b) = a² - b², using (3x + 4)(3x - 4) = 9x² - 16 as an example.

🛎️ Squaring a Binomial: Where the Formula Comes From

  • Squaring a binomial means multiplying it by itself, like (a+b)(a+b)(a+b)(a+b).
  • The middle term appears twice, which is why we get 2ab2ab.

🛎️ Squaring a Binomial: See It Visually

  • The square splits into four parts showing a2a^2, b2b^2, and two abab rectangles.
  • Adding the areas gives (a+b)2=a2+b2+2ab(a+b)^2 = a^2 + b^2 + 2ab.

🛎️ The 3 Key Formulas

  • (a+b)2=a2+b2+2ab(a+b)^2 = a^2 + b^2 + 2ab.
  • (a−b)2=a2+b2−2ab(a-b)^2 = a^2 + b^2 - 2ab.
  • (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.

🛎️ Using (a+b)2(a+b)^2: Example

  • Identify a and b including their signs, for example in (−2x+3)2(-2x+3)^2 we have a = -2x and b = 3.
  • Work out squares first, then add the end term to get (−2x)2+32+2×(−2x)×3=4x2+9−12x(-2x)^2 + 3^2 + 2 \times (-2x) \times 3 = 4x^2 + 9 - 12x.

🛎️ Using (a−b)2(a-b)^2: Example

  • Identify a and b including the minus sign, for example in (5x−2y)2(5x-2y)^2 we have a = 5x and b = 2y.
  • Work out squares first, then subtract the end term to get (5x)2+(2y)2−2×5x×2y=25x2+4y2−20xy(5x)^2 + (2y)^2 - 2 \times 5x \times 2y = 25x^2 + 4y^2 - 20xy.

🛎️ Using (a+b)(a−b)(a+b)(a-b): Example

  • Check the brackets use the same terms with different signs, for example (3x+4)(3x−4)(3x+4)(3x-4).
  • Square both terms and subtract to get (3x)2−42=9x2−16(3x)^2 - 4^2 = 9x^2 - 16.

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