Quadratics

• What are the methods for solving quadratic equations mentioned in the study material?

  Factorising, graphically, quadratic formula, and discriminant
• What does it mean to 'solve' a quadratic equation?

  • To find the roots of the quadratic
  • To determine where the parabola cuts the x-axis
  • Substitute \(y = 0\) into the equation
• How do you solve the quadratic equation \({x^2} - 9x + 20 = 0\)?

  Factorise to \((x - 4)(x - 5) = 0\)
• What are the roots of the equation \({x^2} - 9x + 20 = 0\)?

  4 and 5
• What is the first step to solve the equation \({x^2} + x - 6 = 0\)?

  Factorise the trinomial
• What is the factorization of \({x^2} + x - 6\)?

  \((x - 2)(x + 3)\)
• What are the two possible \(x\) values from the equation \({x^2} + x - 6 = 0\)?

  2 and -3
• What are the key features of quadratic functions that can be identified from their graphs?

  • Vertex
  • Axis of symmetry
  • Roots (x-intercepts)
  • Direction of opening (upward or downward)
• What is the significance of the discriminant in solving quadratic equations?

  • Determines the number of roots
  • If positive, two distinct real roots
  • If zero, one real root
  • If negative, no real roots
• What is the quadratic formula used for?

  • To find the roots of a quadratic equation
  • Given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• What is the process of factorising a quadratic equation?

  1. Rewrite the quadratic in standard form
  2. Find two numbers that multiply to \(c\) and add to \(b\)
  3. Express the quadratic as a product of two binomials
• What is the graphical method of solving quadratic equations?

  • Plot the quadratic function on a graph
  • Identify the points where the graph intersects the x-axis
  • These points are the roots of the equation