Advanced Math • Quadratic Equations and Parabola

Factored Form Expansion

Master the art of expanding quadratics rapidly. Recognize special patterns to save precious time on the SAT Math section.

The FOIL Method

When you multiply two binomials together, you have to distribute every term in the first parenthesis to every term in the second parenthesis. The acronym FOIL helps you remember the order:

F.O.I.L. Breakdown

  • First: Multiply the first terms.
  • Outer: Multiply the outer terms.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms.
\((x + 2)(x + 3)\)
F: \(x \cdot x = x^2\)
O: \(x \cdot 3 = 3x\)
I: \(2 \cdot x = 2x\)
L: \(2 \cdot 3 = 6\)
Result: \(x^2 + 5x + 6\)

Special Quadratic Patterns

The SAT loves to test your knowledge of special multiplication patterns. Memorizing these will allow you to bypass the FOIL method entirely and mentally expand equations in seconds.

Perfect Square Trinomials

A binomial squared always yields a perfect square trinomial. The middle term is twice the product of the terms!
\((a + b)^2 = a^2 + 2ab + b^2\)
\((a - b)^2 = a^2 - 2ab + b^2\)
Example 1:
\((x + 5)^2 = x^2 + 10x + 25\)
Example 2:
\((2x - 3)^2 = 4x^2 - 12x + 9\)

Difference of Squares

When you multiply a binomial by its conjugate (same terms, opposite sign), the middle terms cancel out perfectly.
\((a + b)(a - b) = a^2 - b^2\)
Example 1:
\((x + 7)(x - 7) = x^2 - 49\)
Example 2:
\((3x + 4)(3x - 4) = 9x^2 - 16\)

⚠️ The Classic SAT Trap

The most common algebra mistake on the SAT is distributing the exponent incorrectly across a binomial. An exponent does NOT distribute across addition or subtraction.

❌ WRONG: \((x + 4)^2 = x^2 + 16\)
✅ RIGHT: \((x + 4)^2 = x^2 + 8x + 16\)

Remember: The SAT will always include the "wrong" answer choice (\(x^2 + 16\)) to catch students who forget the middle term!