Quadratic Formula, Knowledge Organiser

Key Terms

• Quadratic formula: A formula that solves any quadratic ax² + bx + c = 0
• Discriminant: b² − 4ac: determines how many solutions exist
• Coefficients a, b, c: From ax² + bx + c = 0 (note: c can be negative)
• Exact answer: An answer in surd form (not a rounded decimal)
• Surd: An irrational square root left in the form √k

Must-Know Facts

• If b² − 4ac > 0: two distinct solutions
• If b² − 4ac = 0: one repeated solution (x = −b ÷ 2a)
• If b² − 4ac < 0: no real solutions
• Always rearrange to ax² + bx + c = 0 BEFORE identifying a, b, c
• The ± gives two solutions: one with + and one with −
• Calculate b² − 4ac first, then substitute carefully
• Good news: this formula IS printed on the Higher tier exam formula sheet, so you don't need to memorise it, but you DO need to know how to substitute a, b and c correctly

Key Formulas

• x = (−b ± √(b² − 4ac)) ÷ 2a
• Discriminant: b² − 4ac
• b² − 4ac > 0 → 2 roots; = 0 → 1 root; < 0 → no real roots

Common Mistakes

• Forgetting the ± : The formula gives TWO solutions: always write both x = (−b + √…)/2a AND x = (−b − √…)/2a
• Dividing only part of the numerator: The entire expression −b ± √(b²−4ac) is divided by 2a, not just the √ part
• Sign of b: If b is negative (e.g. b = −5), then −b = +5: take care with negatives
• Not rearranging first: Must have ax² + bx + c = 0 before identifying a, b, c
• Rounding too early: Keep √(b²−4ac) exact until the final step to maintain accuracy

Worked Examples

Solve: x² + 5x + 6 = 0 using the quadratic formula Step 1 Identify a, b, c a = 1, b = 5, c = 6 Step 2 Calculate b² - 4ac b² - 4ac = 5² - 4(1)(6) = 25 - 24 = 1 Step 3 Apply the formula x = (-5 ± √1) / 2(1) x = (-5 ± 1) / 2 Step 4 Find both solutions x = (-5 + 1) / 2 = -4/2 = -2 x = (-5 - 1) / 2 = -6/2 = -3 Answer: x = -2 or x = -3

Worked Examples

Solve: x² + 4x - 1 = 0, giving answers in surd form Step 1 Identify a, b, c a = 1, b = 4, c = -1 Step 2 Calculate b² - 4ac b² - 4ac = 4² - 4(1)(-1) = 16 + 4 = 20 Step 3 Apply the formula x = (-4 ± √20) / 2 Simplify √20 = √(4×5) = 2√5 x = (-4 ± 2√5) / 2 x = -2 ± √5 Answer: x = -2 + √5 or x = -2 - √5