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Completing the Square
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Foundational skills
Expand a perfect square of the form (x + a)²
\[ (x + 4)^2 \rightarrow x^2 + 8x + 16 \]
Expand brackets of the form (x + a)².
Expand a perfect square of the form (x + a)² + q
\[ (x + 3)^2 – 5 \]
Expand completed square expressions.
Identify the value of p from completed square form
\[ (x + 5)^2 – 7 \text{ : find } p \]
Read off the value of p from (x + p)² + q.
Identify the value of q from completed square form
\[ (x – 2)^2 + 11 \text{ : find } q \]
Read off the value of q from (x + p)² + q.
Calculate half of an even number
\[ 8 \div 2 = \square \]
Find half of an even integer.
Calculate (b/2)² for an even coefficient
\[ b = 6 \text{ : find } \left(\frac{b}{2}\right)^2 \]
Square half of an even coefficient.
Calculate (b/2)² for an odd coefficient
\[ b = 5 \text{ : find } \left(\frac{b}{2}\right)^2 \]
Square half of an odd coefficient.
Completing the square (a = 1)
Complete the square for x² + bx (b even positive)
\[ x^2 + 6x \]
Complete the square with no constant term.
Complete the square for x² – bx (b even positive)
\[ x^2 – 8x \]
Complete with negative coefficient of x.
Complete the square for x² + bx (b odd)
\[ x^2 + 5x \]
Complete with odd coefficient giving fractions.
Complete the square for x² + bx + c (q positive)
\[ x^2 + 4x + 7 \]
Complete with positive constant q.
Complete the square for x² + bx + c (q negative)
\[ x^2 + 6x + 2 \]
Complete with negative constant q.
Complete the square for x² + bx + c (b odd)
\[ x^2 + 3x + 1 \]
Complete with odd b giving fractional q.
Complete the square for x² – bx + c
\[ x^2 – 10x + 20 \]
Complete when x coefficient is negative.
Complete the square for x² + bx – c
\[ x^2 + 8x – 5 \]
Complete when constant term is negative.
Completing the square (a ≠ 1)
Factor out the coefficient of x² (setup step)
\[ 2x^2 + 12x + 7 \rightarrow 2(\ldots) + 7 \]
Factor out a from x² and x terms only.
Complete the square for ax² + bx + c (clean values)
\[ 2x^2 + 8x + 5 \]
Complete with a > 1 and clean arithmetic.
Complete the square for ax² + bx + c (general)
\[ 3x^2 + 5x + 1 \]
Complete with a > 1, may involve fractions.
Complete the square for ax² + bx + c (a negative)
\[ -x^2 + 6x – 2 \]
Complete with negative leading coefficient.
Interpreting completed square form
Find the minimum value from (x + p)² + q
\[ (x + 3)^2 + 5 \text{ : min value} \]
Identify the minimum value of a quadratic.
Find the minimum value from a(x + p)² + q (a > 0)
\[ 2(x – 1)^2 + 7 \text{ : min value} \]
Find minimum when a > 1.
Find the maximum value from a(x + p)² + q (a < 0)
\[ -(x + 2)^2 + 10 \text{ : max value} \]
Find maximum when a < 0.
Find the x-coordinate of the turning point
\[ (x + 4)^2 – 3 \text{ : x-coord} \]
Find x where minimum/maximum occurs.
Find the turning point from (x + p)² + q
\[ (x – 2)^2 + 6 \text{ : turning pt} \]
Find complete coordinate pair.
Find the turning point from a(x + p)² + q
\[ 3(x + 1)^2 – 12 \text{ : turning pt} \]
Find turning point when a ≠ 1.
State whether a quadratic has a min or max
\[ -2(x – 5)^2 + 8 \text{ : min or max?} \]
Determine from the sign of a.
Find the equation of the line of symmetry
\[ (x – 7)^2 + 2 \text{ : line of sym} \]
Find the vertical line of symmetry.
Find the y-intercept from completed square form
\[ (x + 3)^2 – 4 \text{ : y-intercept} \]
Substitute x = 0 to find y-intercept.
Finding turning points by completing the square
Complete the square then find the turning point (a = 1)
\[ x^2 + 6x + 5 \text{ : turning pt} \]
Complete and extract turning point.
Complete the square then find the turning point (a ≠ 1)
\[ 2x^2 + 8x + 3 \text{ : turning pt} \]
Complete and find turning point for a ≠ 1.
Find the min or max value by completing the square
\[ x^2 – 4x + 9 \text{ : min value} \]
Complete to find extreme value.
Find the line of symmetry by completing the square
\[ x^2 + 10x + 21 \text{ : line of sym} \]
Complete to find axis of symmetry.
Working backwards
Write a quadratic given the minimum point
\[ \text{Min at } (3, -2) \rightarrow (x+p)^2+q \]
Construct completed square from minimum.
Write a quadratic given the maximum point
\[ \text{Max at } (-1, 5) \rightarrow -(x+p)^2+q \]
Construct completed square from maximum.
Write a quadratic given turning point and coefficient
\[ \text{TP } (2,-7), a=3 \rightarrow a(x+p)^2+q \]
Construct with given a value.
Special cases
Recognise a perfect square quadratic
\[ x^2 + 12x + 36 \rightarrow (x+6)^2 \]
Complete when q = 0.
Complete the square giving q = 0
\[ x^2 – 14x + 49 \]
Recognise c = (b/2)² exactly.
Timer (Optional)
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