Solving quadratic equations

Quadratic equations can be solved by the following methods:

  1. factorising
  2. graphically
  3. quadratic formula
  4. discriminant

Factorising

Look at the National 4 factorising section before continuing.

When a question asks you to 'solve' a quadratic equation, this means that you are to find the roots of the quadratic. In other words, where does the parabola cut the x-axis?

As a graph cuts the axis when the y coordinate is zero, then we substitute \(y = 0\) into the quadratic equation and use algebra to solve.

Example

Solve \({x^2} - 9x + 20 = 0\)

We need to factorise the trinomial.

When factorised this is \((x - 4)(x - 5) = 0\).

\((x - 4)\) and \((x - 5)\) are multiplying to give zero, therefore one of these brackets must be equal to zero.

\[(x - 4) = 0\]

\[x = 0 + 4\]

\[x = 4\]

and

\[(x - 5) = 0\]

\[x = 0 + 5\]

\[x = 5\]

Therefore \(x = 4\,and\,x = 5\) are the roots of quadtratic equations.

Now try the example question below.

Question

Solve \({x^2} + x - 6\)

Factorise the trinomial and then find the two possible \(x\) values.

\[{x^2} + x - 6 = 0\]

\[(x - 2)(x + 3) = 0\]

\[(x - 2) = 0\,and\,(x + 3) = 0\]

\[x = 0 + 2 = 2\,and\,x = 0 - 3 = - 3\]

Therefore \(x = 2\,and\,x = 3\)