What you’ll learn
Rewrite quadratic expressions by completing square.
Key idea
The grouped square contributes its own constant; subtract that contribution outside to preserve the original expression.
Halve the signed linear coefficient and compensate outside for the square contribution; preserve the polynomial exactly.
Worked example
Enter the exact value or ordered values in the format requested below.
Supplied quadratic quantity x²-16x+5 must be represented as (x+a)²+b. Enter the completed-square expression, not its expansion.
- Half the signed linear coefficient -16 is -8; its square contributes 64.
- Compensate outside: 5-64=-59, yielding (x-8)²-59. Vertex input 8 zeros the square, with minimum -59.
Answer: (x-8)²-59
Quick Check
Choose which candidate representation satisfies the requirements of 'Rewrite quadratic expressions by completing square' and justify the choice.
An upward-opening monic quadratic has vertex (7,-40) and expanded expression x²-14x+9. Select the matching completed-square representation and justify both constraints.