Completing the square (advanced)
Completing the Square
Completing the Square is the opening topic of Quadratic Functions in the California Common Core State Standards. It is aligned to Standard A-REI.4a, which requires students to solve quadratic equations by inspection, completing the square, and the quadratic formula.
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.