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Complete the square and link equivalent forms

Start a set of 8
Step 1 of 6
Height rule
\(\displaystyle h(x)=2x^2-16x+35\)
Horizontal interval
\(\displaystyle 0\le x\le 8\)

A curved path has the height rule shown, where x is the horizontal distance in metres and height is measured in metres. At what horizontal position is the lowest point of the path?

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

Complete the square

For a quadratic with real coefficients and a nonzero leading coefficient, the displayed form has vertex input coordinate h and output coordinate k. The leading coefficient multiplies the whole square: complete the monic bracket, then multiply the correction by a

\[f(x)=a(x-h)^2+k\]
  1. Factor the variable terms Take the leading coefficient outside the terms containing the variable before completing the square \(\displaystyle 2x^2+12x+5=2(x^2+6x)+5\)
  2. Halve the linear coefficient Inside the monic bracket, halve the coefficient of x and use that value inside the square \(\displaystyle x^2+6x=(x+3)^2-9\)
  3. Square the correction The number subtracted outside the square is the square of the half coefficient, not the half coefficient itself \(\displaystyle 3^2=9\)
  4. Read the vertex In the displayed form, the vertex has input coordinate h and output coordinate k \(\displaystyle 2(x+3)^2-13\Rightarrow\text{vertex }(-3,-13)\)

Worked example

Express the displayed rule in completed-square form

\[f(x)=3x^2-18x+7\]
  1. \(\displaystyle 3x^2-18x+7=3(x^2-6x)+7\)Factor out the leading coefficient from the variable terms
  2. \(\displaystyle x^2-6x=(x-3)^2-9\)Half the coefficient of x is -3, and its square is 9.
  3. \(\displaystyle 3\left[(x-3)^2-9\right]+7=3(x-3)^2-27+7\)Distribute the leading coefficient through the completed bracket
  4. \(\displaystyle f(x)=3(x-3)^2-20\)Collect the outside constants to obtain the completed-square form

Common mistakes

Using the half coefficient unsquared
✕\(\displaystyle x^2+10x=(x+5)^2-5\)✓\(\displaystyle x^2+10x=(x+5)^2-25\)
The correction is the square of the half coefficient, so 5 must be squared to give 25.
Dropping the leading factor
✕\(\displaystyle 2x^2+12x+1=(x+3)^2-8\)✓\(\displaystyle 2x^2+12x+1=2(x+3)^2-17\)
The leading coefficient multiplies the completed square and must remain attached when the constant is collected
Reversing the horizontal sign for vertex (4,−5)
✕\(\displaystyle f(x)=2(x+4)^2-5\)✓\(\displaystyle f(x)=2(x-4)^2-5\)
For the stated vertex, subtract 4 inside the bracket. Adding 4 would put the vertex at a horizontal coordinate of negative 4

Difficulty levels

introComplete monic quadratics with integral translations from general form
standardConvert in either direction with a non-unit leading scale and exact rational translations; identify the vertex coordinate or extremum
challengeConvert in either direction with positive or negative rational scales, retaining exact tuple and vertex values
examSelect the equivalent form, enter the two missing tuple values and find the remaining vertex coordinate or extremum

Questions about this topic

Why is the leading coefficient factored out first?+

It makes the bracket monic, so the half-coefficient rule applies correctly. The factor must stay outside the completed square

Why is the half coefficient squared?+

The correction must be the square of the half coefficient to keep the expression equivalent

How is the vertex read from completed-square form?+

In completed-square form, the vertex coordinates are h and k. Read the bracket sign carefully: adding 4 inside it gives a horizontal vertex coordinate of negative 4

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 3: Quadratic functions

In this chapter

Factorise, solve and reconstruct a rational quadraticCalculate and interpret a quadratic discriminantTransform a quadratic and find its key featuresComplete the square and link equivalent formsDetermine a quadratic rule from exact informationSolve a quadratic exactly with the quadratic formulaConstruct a rational quadratic from a surd rootSolve an equation that reduces to quadratic formAnalyse signs and solve a quadratic inequalityIntersections of a line and a quadratic graphDetermine parameter values from a discriminant conditionFormulate and solve a quadratic problemAnalyse and optimise a quadratic model