Grade 9

Quadratic Equations

Factorising, the quadratic formula, the discriminant, and the parabola that all three of them describe.

6 lessons

  1. 1How do you expand brackets and what are the special products?Multiply every term in the first bracket by every term in the second, then collect like terms. Three formulas shortcut the common cases: the square of a sum, the square of a difference, and the difference of two squares. They are not extra rules but results of the same expansion, worth recognising on sight.35m
  2. 2How do you factorise an algebraic expression?Factorising is expanding brackets in reverse: it turns a sum into a product. Always try in the same order — common factor first, then a special product formula, and only if neither fits, factorise the trinomial by finding two numbers whose product is the constant and whose sum is the middle coefficient.45m
  3. 3How do you solve a quadratic equation by factorising?Move every term to one side so the other side is zero, factorise the expression into a product, then use the fact that a product equals zero only if one of its factors is zero. Each factor is set to zero separately, so one quadratic equation splits into two simple ones.35m
  4. 4What is the quadratic formula and how do you use it?The quadratic formula solves any quadratic equation from its three coefficients alone: minus the middle coefficient, plus or minus the square root of the middle coefficient squared minus four times the leading coefficient times the constant, all divided by twice the leading coefficient. It works even when the expression does not factorise.40m
  5. 5What is the discriminant and how many solutions does an equation have?The discriminant is the expression under the root in the quadratic formula, and it alone decides the number of solutions. Positive means two different solutions, zero means exactly one, and negative means no real solution. You can determine this without solving the equation at all.30m
  6. 6How do you find the vertex and the intercepts of a parabola?The intercepts with the horizontal axis are exactly the solutions of the quadratic equation, and the intercept with the vertical axis is the constant term. The x-coordinate of the vertex is minus the middle coefficient divided by twice the leading one, and substituting it gives the y-coordinate. The sign of the leading coefficient says which way the parabola opens.45m