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Rewrite a quadratic as a squared bracket plus a number. Halving the x coefficient gives the number inside the bracket, and subtracting its square fixes the total. The completed form hands you the turning point of the graph and the smallest value the expression can take, which is what exam questions ask for next.

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    Worked Example

    Write x² + 6x + 11 in completed square form.

    1. Halve the number with x:6 ÷ 2 = 3, so the bracket is (x + 3)
    2. Squaring that bracket adds an extra 9, so subtract it again
    3. Tidy the numbers:11 − 9 = 2, so x² + 6x + 11 = (x + 3)² + 2

    Answer: (x + 3)² + 2

    Try These Yourself

    Write x² + 4x + 7 in completed square form.Show answerHide answer

    Answer: (x + 2)² + 3

    Half of 4 is 2, and 7 − 4 = 3.

    Write x² − 10x + 30 in completed square form.Show answerHide answer

    Answer: (x − 5)² + 5

    Half of −10 is −5, and 30 − 25 = 5.

    What is the turning point of (x + 3)² + 2?Show answerHide answer

    Answer: (−3, 2)

    The bracket is zero when x = −3.