Quadratic Equations

Understanding Quadratic Equations

  • A quadratic equation is an equation that can be written in the standard form ( ax^2 + bx + c = 0 ), where ‘a’, ‘b’, and ‘c’ are constants, ‘x’ is the variable, and ( a ≠ 0 ).
  • The highest power of the variable in a quadratic equation is two. So if an equation involves a variable squared but not cubed, it is likely to be a quadratic equation.
  • Quadratic equations are used to solve problems involving area, speed, motion, trajectory, and many others in real life.

Solving Quadratic Equations

  • The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle is often used to solve quadratic equations.
  • To solve a quadratic equation in the standard form, you can either factorise it, complete the square, or use the quadratic formula (for Higher Tier).
  • Quadratic equations that can be written in the form ( (x-a)^2 = 0 ) can be solved easily by taking the square root of both sides of the equation.

Factoring Quadratic Equations

  • Quadratic equations of the form ( x^2 - c^2 = 0 ) can often be factored into ( (x+c)(x-c) = 0 ).
  • The coefficients in the quadratic equation can provide clues about how to factor.
  • Knowing your multiplication table will enable you to factorise easily.

Graphs of Quadratic Equations

  • Quadratics are best represented graphically via a parabola, which is a U-shaped curve that may open upwards (if a>0) or downwards (if a<0).
  • The vertex of the parabola is the minimum or maximum point of the parabola. It is given by the formula (-b/2a, f(-b/2a)) where f is the quadratic function.
  • The axis of symmetry of a parabola is the vertical line that passes through the vertex, and it has the equation x = -b/2a.

Common Mistakes with Quadratic Equations

  • Misinterpretation of the standard form can lead to errors in calculation. Always make sure that your quadratic is in the correct order ( ax^2 + bx + c = 0 ) before solving.
  • If you’re factoring, lack of knowledge about the multiplication table can lead to incorrect factor pairs. Factor pair knowledge should be well-revised.
  • Incorrect graph plotting or reading off the solutions from graphs incorrectly is a common mistake. Always ensure to read the graph accurately.