Completing the square (advanced)
Completing the Square
Completing the Square is the opening topic of Quadratic Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.4(D), which requires students to transform a quadratic function into f(x)=a(x-h)^2+k form by completing the square.
Completing the square turns a quadratic into vertex form \(a(x-h)^2+k\) by adding \(\left(\dfrac{b}{2}\right)^2\).
Theory
This produces vertex form \(a(x-h)^2+k\), where \((h,k)\) is the vertex.
The key step:
How to complete the square
- If \(a\neq1\), factor \(a\) from the \(x^2\) and \(x\) terms.
- Take half of \(b\) and square it.
- Add and subtract that value to form a perfect square.
- Write as \(a(x-h)^2+k\).
Half of \(6\) is \(3\); add and subtract \(3^2=9\).
| \((x^2+6x+9)-9+5\) | ||
| \(=\) | \((x+3)^2-4\) |
Move the constant, then complete the square.
| \(x^2-4x\) | \(=\) | \(1\) |
| \(x^2-4x+4\) | \(=\) | \(5\) |
| \((x-2)^2\) | \(=\) | \(5\) |
| \(x\) | \(=\) | \(2\pm\sqrt5\) |
Complete the square: half of \(8\) is \(4\).
| \((x^2+8x+16)-16+10\) | ||
| \(y\) | \(=\) | \((x+4)^2-6\) |
The vertex is \((-4,-6)\).
Factor \(2\) from the \(x\)-terms first.
| \(2(x^2+4x)+3\) | ||
| \(=\) | \(2(x^2+4x+4)-8+3\) | |
| \(=\) | \(2(x+2)^2-5\) |
Common pitfalls
Frequently asked questions
What is completing the square?
Adding \((b/2)^2\) to make \(x^2+bx\) a perfect square trinomial.
What is vertex form?
\(a(x-h)^2+k\), where \((h,k)\) is the vertex.
What if the leading coefficient isn't \(1\)?
Factor it out of the \(x\)-terms before completing the square.
Why add and subtract the same value?
So the expression stays equal while forming the perfect square.