Math: Completing the Square

Understanding Math: Completing the Square

The reason the method uses half of the middle coefficient comes from multiplication. Squaring a binomial means multiplying it by itself. The first term produces a squared variable term.

The last term produces a number squared. The two cross products are equal, so together they make twice the number beside the variable.

This pattern is worth memorising because it lets students recognise a hidden square instead of relying on a rule with no explanation. For example, a middle term of ten means the number inside the binomial must be five, since twice five is ten.

A leading coefficient changes the bookkeeping. Consider three times the variable squared minus twelve times the variable plus seven. The first two terms share a factor of three.

Inside the brackets, the variable squared minus four times the variable needs four to become a square, because half of negative four is negative two. Adding four inside brackets really adds twelve to the original expression, since every term in the brackets is multiplied by three. That extra twelve must be removed outside the brackets.

After simplifying, the expression becomes three times the square of the quantity variable minus two, minus five. The vertex can now be read directly as two units right and five units down.

This form is especially useful when an equation is set equal to zero. Once the squared part has been isolated, the next step is to find numbers whose square gives the remaining value. A positive remaining value produces two real solutions, placed equally far to either side of the vertex line.

A remaining value of zero produces one repeated solution at the vertex. A negative remaining value has no real solution because no real number squares to a negative value.

This explains a feature that can seem mysterious in quadratic graphs. A parabola can cross the horizontal axis twice, touch it once, or miss it completely.

Students meet this idea whenever a quantity rises then falls, or falls then rises. A ball thrown upward has a height that can be written in a form showing its greatest height and the time when it occurs. A business model may use a quadratic to represent profit, where the vertex gives the highest or lowest possible value under the model.

The main mistakes are usually small arithmetic errors. Keep track of signs when halving the middle coefficient. Square the result before using it.

When a number has been factored from the variable terms, remember that the correction is affected by that outside factor. Finally, expand the completed form once as a check. If it does not return to the original expression, one of the balancing steps changed the value.