Quadratic Equations

Summary and Exam Tips for Quadratic Equations

Quadratic Equations is a subtopic of Algebra, which falls under the subject Mathematics in the IB MYP curriculum. When solving quadratic equations, we typically find two different solutions. There are two primary methods to solve these equations: factorization and using the quadratic formula.

1. Factorization Method:

   • Rearrange the equation to set it equal to zero, e.g., $x^2 - x - 6 = 0$.
   • Factorize the quadratic equation, e.g., $(x-3)(x+2) = 0$.
   • Set each factor equal to zero to find the solutions, $x = 3$ or $x = -2$.

2. Algebraic Manipulation:

   • For equations like $16x^2 - 9 = 0$, rearrange to $x^2 = \frac{9}{16}$.
   • Solve for $x$ to get $x = \frac{3}{4}$ or $x = -\frac{3}{4}$.

3. Quadratic Formula:

   • Use when factorization is not possible. For $ax^2 + bx + c = 0$, apply the formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
   • Example: Solve $24x^2 - 22x - 35 = 0$ using $a = 24$, $b = -22$, $c = -35$.

Remember: Always provide both positive and negative solutions when applicable, and only use square roots on positive numbers.

Exam Tips

• Understand the Methods: Familiarize yourself with both factorization and the quadratic formula. Knowing when to use each method is crucial.
• Practice Factorization: Ensure you can quickly factorize simple quadratics, as this is often the fastest method.
• Check Your Solutions: Always substitute your solutions back into the original equation to verify their correctness.
• Memorize the Formula: The quadratic formula is essential for solving non-factorable equations, so commit it to memory.
• Work on Word Problems: Practice translating word problems into quadratic equations, as this is a common exam scenario.