Skip to content

Quadratic equations

See also: mathematical-content-knowledge, teaching-mathematics

Definitions#

A quadratic is an expression of the form \(ax^2 + bx + c\) where \(a, b, c\) are constants and \(a \neq 0\).

Why?#

Related to linear equations where solving a linear equation focuses on isolating the unknown. Making it easier to identify the solution.

Isolating the unknown doesn't work for quadratics

Quadratics aren't overly obvious in everyday life, but are used in advanced mathematics.

Quadratics crop up when solving complicated/important real world problems

Declarative knowledge#

A quadratic is an expression of the form \(ax^2 + bx + c\) where \(a, b, c\) are constants and \(a \neq 0\).

The standard form of a quadratic equation takes the form

\(ax^2 + bx + c = 0,\) where \(a, b, c\) are constants and \(a \neq 0\)$.

The aim is to find the value of \(x\) that makes the equation true. These are the solutions to the equation.

Quadratic equations can have 0, 1 or 2 solutions.

The constants \(a, b, c\) are also known as the coefficients.

Procedural knowledge#

Simplifying quadratic expressions#

Prior to solving a quadratic equation, it is often useful to simplify the expression by

  • Remove a negative \(a\) by multiplying through by \(-1\)
  • Remove any fractions for \(a\) and \(b\) by multiplying through by the lowest common denominator
  • Reduce by any common factor to make the coefficients as small as possible
  • Rewriting equations that are not in the standard form

Solving variations of quadratic equations#

Quadratic equations with no term in \(x\)#

Can be done with simple isolation - similar to linear equations

Quadratic with no constant term#

One of the solutions will be \(x = 0\) and the other will be the solution to the linear equation \(ax + b = 0\).

Solving quadratic equations with three terms#

Three basic methods

  1. Factorisation
  2. Completing the square
  3. Quadratic formula

Factorisation#

Related to the method used to the use of the zero product rule to help solve the "no constant" version.

When \(a = 1\) then search for the factors of \(c\) that add to \(b\).

if \(a \neq 1\) then find the factors of \(ac\) that add to \(b\)

One way to check if a quadratic can be factorised is to use the discriminant. If the discriminant is a perfect square then the quadratic can be factorised. The discriminant is given by the formula \(b^2 - 4ac\).

Cross method#

A method for factorising quadratics that is often taught in schools. It is a method that is easy to teach and easy to learn. It is not a method that is used elsewhere in mathematics.

Form a x by

  • Choose 2 factors of \(a\) and place them in the top left and bottom left corners (include the x)
  • Choose 2 factors of \(c\) and place them in the top right and bottom right corners
  • Multiply the factors across the x
  • top left x bottom right
  • bottom left x top right
  • Add the two products together to test if the result is \(b\)
  • Write the factors from each row and then solve....there's more here

Completing the square#

A more general method, especially when there are no common factors. A method that is used for other mathematical purposes. Will be needed elsewhere.

lots more here

Quadratic formula#

Always works, by entering the values of \(a, b, c\) into the formula

\[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The discriminant is given by the formula \(b^2 - 4ac\) and can be used to identify if there are solutions:

  • If the discriminant is positive then there are two solutions
  • If the discriminant is zero then there is one solution
  • If the discriminant is negative then there are no solutions

Resources#