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Solving Quadratic Equations

SAT MathAdvanced MathMedium

Choose factoring, square roots, completing the square, or the quadratic formula efficiently.

What you need to know

  • Set the equation equal to zero before using the zero-product property.
  • Factoring is fastest when integer factors are visible.
  • For (x − h)² = k, take both positive and negative square roots.
  • The discriminant b² − 4ac indicates the number of real solutions.

The AYLOTI method

  1. Write standard form ax² + bx + c = 0.
  2. Try factoring briefly.
  3. Use square-root form or the quadratic formula when factoring is inefficient.
  4. Check for extraneous or omitted roots.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

Worked example

Solve x² − 5x − 14 = 0.

  1. Find factors with product −14 and sum −5: −7 and 2.
  2. Factor: (x − 7)(x + 2) = 0.
  3. Set each factor to zero.

Answer: x = 7 or x = −2.

Common SAT traps

  • Forgetting to set the equation to zero.
  • Taking only the positive square root.
  • Using b instead of −b in the formula.
Quick check

Solve (x − 4)² = 9.

x = 1 or x = 7.

Before you move on

  • Explain the core rule without looking back.
  • Redo the worked example using your own steps.
  • Write the trap you are most likely to fall for.
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