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Solving Quadratics by Completing the Square
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Foundational skills
Solve (x + a)² = b with integer solutions
\[ (x + 3)^2 = 16 \]
Solve completed square form where √b is an integer.
Solve (x + a)² = b with surd solutions
\[ (x + 2)^2 = 5 \]
Solve giving solutions in surd form.
Solve (x – a)² = b with surd solutions
\[ (x – 4)^2 = 7 \]
Completed square with subtraction in bracket.
Solve (x + a)² = b giving answers to 1 d.p.
\[ (x + 1)^2 = 10 \]
Decimal answers rounded to 1 decimal place.
Solve (x – a)² = b giving answers to 2 d.p.
\[ (x – 3)^2 = 15 \]
Decimal answers rounded to 2 decimal places.
Recognise no real solutions
\[ (x + 2)^2 = -5 \]
A square cannot equal a negative.
Monic quadratics with even x-coefficient
Solve x² + bx + c = 0 (b even) integer solutions
\[ x^2 + 6x + 5 = 0 \]
Even x-coefficient giving integer solutions.
Solve x² + bx + c = 0 (b even) surd solutions
\[ x^2 + 4x – 1 = 0 \]
Even x-coefficient giving surd solutions.
Solve x² + bx + c = 0 (b even) decimal answers
\[ x^2 + 8x + 3 = 0 \]
Decimal answers to 2 d.p.
Solve x² – bx + c = 0 (b even positive)
\[ x^2 – 10x + 21 = 0 \]
Negative x-coefficient.
Monic quadratics with odd x-coefficient
Solve x² + bx + c = 0 (b odd) surd solutions
\[ x^2 + 5x + 2 = 0 \]
Odd x-coefficient requiring fractions.
Solve x² + bx + c = 0 (b odd) decimal answers
\[ x^2 + 7x + 5 = 0 \]
Odd x-coefficient with decimal answers.
Monic quadratics requiring rearrangement
Solve x² + bx = c
\[ x^2 + 6x = 7 \]
Constant on the right side.
Solve x² + c = bx
\[ x^2 + 10 = 7x \]
x-term on wrong side.
Solve x² = bx + c
\[ x^2 = 4x + 5 \]
x² isolated on one side.
Expand x(x + a) = b then solve
\[ x(x + 6) = 16 \]
Expand first, then complete the square.
Non-monic quadratics
Solve ax² + bx + c = 0 (a = 2) exact solutions
\[ 2x^2 + 8x + 3 = 0 \]
Divide by 2 first.
Solve ax² + bx + c = 0 (a = 2) decimal answers
\[ 2x^2 + 10x + 7 = 0 \]
a = 2, decimal answers to 2 d.p.
Solve ax² + bx + c = 0 (a = 3) exact solutions
\[ 3x^2 + 12x + 7 = 0 \]
Divide by 3 first.
Solve ax² + bx + c = 0 (a ≥ 3) decimal answers
\[ 4x^2 + 12x + 5 = 0 \]
Larger leading coefficient, decimal answers.
Solve with negative leading coefficient
\[ -x^2 + 6x – 4 = 0 \]
Multiply by -1 first.
Non-monic requiring rearrangement
Solve ax² + bx = c
\[ 2x^2 + 6x = 5 \]
Non-monic with constant on wrong side.
Expand ax(x + b) = c then solve
\[ 2x(x + 3) = 7 \]
Expand, divide, then complete square.
Special cases
Quadratic with no real solutions
\[ x^2 + 4x + 6 = 0 \]
Completing square gives negative.
Quadratic with repeated root
\[ x^2 + 8x + 16 = 0 \]
Perfect square giving one solution.
Solve x² – c = 0 (no x-term)
\[ x^2 – 9 = 0 \]
No x-term to complete.
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