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Solving Quadratics Using the Formula
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Foundational skills
Identify a, b and c from standard form
\[ 3x^2 + 5x – 2 = 0 \]
Identify coefficients from ax² + bx + c = 0
Identify a, b, c when coefficient is 1
\[ x^2 + 7x – 3 = 0 \]
Identify coefficients when x² coefficient is implicit
Identify a, b, c with negatives
\[ 2x^2 – 3x – 7 = 0 \]
Identify coefficients when some are negative
Identify a, b, c when term is missing
\[ 5x^2 – 20 = 0 \]
Identify coefficients when b = 0 or c = 0
Standard form with integer solutions
Monic quadratic with integer solutions
\[ x^2 – 5x + 6 = 0 \]
Use the formula on x² + bx + c = 0 with integer solutions
Non-monic quadratic with integer solutions
\[ 2x^2 – 7x + 3 = 0 \]
Use the formula on ax² + bx + c = 0 with integer solutions
Standard form with surd solutions (exact)
Monic quadratic with surd solutions
\[ x^2 + 4x + 1 = 0 \]
Solve x² + bx + c = 0 giving exact surd answers
Non-monic quadratic with surd solutions
\[ 2x^2 + 6x + 1 = 0 \]
Solve ax² + bx + c = 0 (a > 1) giving exact surd answers
Simplify surd solutions fully
\[ x^2 + 8x + 4 = 0 \]
Solve quadratic requiring full surd simplification
Standard form with decimal solutions
Monic quadratic to 1 d.p.
\[ x^2 + 3x – 5 = 0 \]
Solve x² + bx + c = 0 to 1 decimal place
Monic quadratic to 2 d.p.
\[ x^2 – 5x + 2 = 0 \]
Solve x² + bx + c = 0 to 2 decimal places
Non-monic quadratic to 2 d.p.
\[ 3x^2 + 7x – 2 = 0 \]
Solve ax² + bx + c = 0 (a > 1) to 2 decimal places
Non-monic quadratic to 3 d.p.
\[ 2x^2 – 5x – 4 = 0 \]
Solve ax² + bx + c = 0 to 3 decimal places
Requiring rearrangement (collect terms)
Solve x² + bx = c
\[ x^2 + 5x = 14 \]
Rearrange then apply the formula
Solve x² = bx + c
\[ x^2 = 7x + 18 \]
Rearrange when x² is isolated
Solve ax² + bx = c
\[ 2x^2 + 3x = 5 \]
Rearrange non-monic quadratic to standard form
Solve ax² + c = bx
\[ 3x^2 + 4 = 8x \]
Rearrange when x-term is on wrong side
Requiring rearrangement (expand brackets)
Expand x(x + a) = b then solve
\[ x(x – 4) = 12 \]
Expand a bracket, rearrange, then apply formula
Expand (x + a)(x + b) = c then solve
\[ (x + 2)(x – 5) = 8 \]
Expand two brackets, rearrange, then apply formula
Expand x(ax + b) = c then solve
\[ x(2x + 5) = 3 \]
Expand non-monic bracket, rearrange, then apply formula
Expand (ax + b)(cx + d) = e then solve
\[ (2x + 1)(x – 3) = 5 \]
Expand two non-monic brackets, then apply formula
Special structures
Solve ax² + c = 0 (no x-term)
\[ 2x^2 – 18 = 0 \]
Apply the formula when b = 0
Solve ax² + bx = 0 (no constant)
\[ 3x^2 + 6x = 0 \]
Apply the formula when c = 0
Solve -x² + bx + c = 0
\[ -x^2 + 7x + 8 = 0 \]
Apply the formula with negative leading coefficient
Solve -ax² + bx + c = 0 where a > 1
\[ -2x^2 + 5x + 12 = 0 \]
Apply the formula with negative leading coefficient > 1
Special cases
Repeated root (discriminant = 0)
\[ x^2 – 6x + 9 = 0 \]
Recognise when the formula gives a repeated root
No real solutions (discriminant < 0)
\[ x^2 + 2x + 5 = 0 \]
Recognise when there are no real solutions
Determine number of solutions from discriminant
\[ 2x^2 + 5x + 4 = 0 \]
Calculate discriminant to find how many real solutions
Hidden quadratics
Solve x⁴ + bx² + c = 0 using substitution
\[ x^4 – 5x^2 + 4 = 0 \]
Substitute u = x² to convert quartic to quadratic
Hidden quadratic with rejected negative u
\[ x^4 + 2x^2 – 8 = 0 \]
Solve where one u-value is negative and must be rejected
Solve x⁶ + bx³ + c = 0 using substitution
\[ x^6 – 9x^3 + 8 = 0 \]
Substitute u = x³ to convert degree-6 to quadratic
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