Quadratic Equations

Understanding Quadratic Equations

Overview

  • A quadratic equation is any equation that can be rearranged in the form ax² + bx + c = 0 where a, b, and c are constants, and x is an unknown variable.
  • The largest power in a quadratic equation is always 2, hence the term ‘quadratic’, derived from ‘quad’ meaning square.
  • Quadratic equations represent parabolic graphs which have a shape of a U or an inverted U.

Solving Quadratic Equations

  • Quadratic equations are solved by factoring, by using the quadratic formula, completing the square or by graphing.
  • The factoring method involves expressing the equation in the form (px + q)(rx + s) = 0 and setting each factor equal to zero, then solving for x.
  • When factoring is not possible, the quadratic formula can be used, which is x = [-b ± sqrt(b² - 4ac)] / 2a.
  • The square root part of the quadratic formula, b² - 4ac, is known as the discriminant. It determines the nature of the roots or solutions of the quadratic equation.

Roots of Quadratic Equations

  • If the discriminant is positive, there are two different real roots.
  • If the discriminant is zero, there is exactly one real root – this is known as a repeated root.
  • If the discriminant is negative, there are two complex roots, and no real roots.

Quadratic Graphs

  • The graph of a quadratic equation is a parabola, which can open upwards or downwards.
  • The vertex of the parabola (the highest or lowest point) represents the maximum or minimum value of the function.
  • The point where the graph cuts the y-axis is called the y-intercept, and points where it cuts the x-axis are roots or solutions of the quadratic equation.

Quadratic Inequalities

  • Quadratic inequalities look similar to quadratic equations but they have an inequality sign (<, >, ≤, or ≥) instead of an equals sign.
  • They are solved in a similar manner to quadratic equations, with the solutions showing a range of values for which the inequality holds true.

Practising Quadratics

  • Regular practice is essential to understand concepts associated with quadratic equations.
  • Try solving different types of problems involving quadratics, and always check your solutions for accuracy.
  • Understanding quadratics is not only beneficial for Algebra, but also critical in various scientific, technological and mathematical disciplines.