Equivalent quadratic forms
Different forms can describe exactly the same function. Choose the form that makes the required feature clear
- Turning-point form For nonzero a, read vertex (h,k) and axis x=h.
- Expanded form The constant is the y-intercept. It is not generally the vertex height
- Complete a square Make the x² coefficient 1 inside the bracket, halve the linear coefficient there, then add and subtract its square
- Check equivalence Expand to prove equality. Use matching graphs and table values to check your working
Worked example
Rewrite in turning-point form and state the vertex
- \(\displaystyle \begin{aligned}y&=x^2-10x+25\\&\quad-25+13\end{aligned}\)Half of −10 is −5; its square is 25. Add and subtract 25.
- \(\displaystyle y=(x-5)^2-12\)Group the perfect square and combine the constants
- \(\displaystyle V=(5,-12)\)The square is zero at x=5.
- \(\displaystyle \begin{gathered}(x-5)^2-12\\=x^2-10x+13\end{gathered}\)Expand back to check the original rule
Common mistakes
Difficulty levels
Questions about this topic
Why is this called completing the square?+
We form a perfect-square bracket by adding and subtracting the same amount
Can I read the vertex from the expanded constant?+
Usually not. The expanded constant gives the y-intercept, which is the value at x=0.
Why halve the linear coefficient?+
The middle term of (x+p)² is 2px. Matching that term gives p as half its coefficient
What if the leading coefficient is not 1?+
Factor it out of the x² and x terms first. Apply any square-completion correction inside that factor
Can h or k be zero?+
Yes. The graph can have its vertex on either axis or at the origin
Do matching graphs prove the algebra?+
A displayed graph is useful evidence, but its finite resolution can hide differences. Expansion proves the identity for every x.
How is this different from solving an equation?+
Rewriting a rule gives an equivalent expression for every input. Solving an equation finds only inputs satisfying a particular equality
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 10: Quadratic equations and graphs