IB Maths AA SL Topic 1 — Number & Algebra Paper 1 & 2 🎯 Skill ~4 min practice

AA SL Solving Quadratic Equations Skills

Three reliable methods, one decision: factor when it’s clean, formula when it’s messy, complete the square when the question demands it. Master the choice and you’ll never get stuck on a quadratic again.

The Method

Standard form: ax² + bx + c = 0

Method 1

Factorising

(x − p)(x − q) = 0

when factors are obvious

Method 2

Quadratic Formula

x = −b ± √(b² − 4ac)2a

always works · in booklet

Method 3

Complete the Square

(x − h)² = k

when “exact form” asked

Which method should I use?

Try factorising first. Look for two numbers that multiply to ac and add to b. If they exist as small integers — factor it. Done in 10 seconds.
If factors are messy or non-integer, switch to the quadratic formula. It works on every quadratic, every time. The discriminant b² − 4ac tells you straight away if there are real solutions.
If the question says “in the form (x − h)² = k” or asks for the turning point or a proof — use complete the square. Don’t skip it; the form is the answer.

Worked examples

WE 1 EASY

Factorising — solve x² + 5x + 6 = 0

step 1 — find the factors need two numbers: × = 6 + = 5 2 and 3 work ✓step 2 — write factored form (x + 2)(x + 3) = 0step 3 — solve each bracket x + 2 = 0 → x = −2 x + 3 = 0 → x = −3x = −2 or x = −3 factor first — 10 seconds when it works!

WE 2 MEDIUM

Quadratic formula — solve 2x² − 7x + 3 = 0

step 1 — identify a, b, c a = 2, b = −7, c = 3step 2 — discriminant b² − 4ac = 49 − 24 = 25step 3 — apply formula x = (7 ± √25) / 4 = (7 ± 5) / 4 x = 12/4 = 3 or x = 2/4 = ½x = 3 or x = ½ if the discriminant is a perfect square → factors existed (could’ve factored)!

WE 3 HARD

Complete the square — solve x² − 6x + 2 = 0, giving exact answers

step 1 — half the b coefficient half of −6 is −3 x² − 6x = (x − 3)² − 9step 2 — rewrite the equation (x − 3)² − 9 + 2 = 0 (x − 3)² = 7step 3 — square root both sides x − 3 = ±√7 x = 3 ± √7x = 3 + √7 or x = 3 − √7 “exact form” means leave the surd — never decimal!

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. Pick the fastest method for each — you’ll see all three appear.

Q1 EASY Solve x² − 7x + 12 = 0 Show answer ▼Hide answer ▲

factor — × = 12, + = −7 −3 and −4 work (x − 3)(x − 4) = 0 x = 3 or x = 4

Q2 EASY Solve x² + 2x − 15 = 0 Show answer ▼Hide answer ▲

factor — × = −15, + = 2 5 and −3 work (x + 5)(x − 3) = 0 x = −5 or x = 3

Q3 MEDIUM Solve 3x² + 5x − 2 = 0 Show answer ▼Hide answer ▲

try formula — a=3, b=5, c=−2 b² − 4ac = 25 + 24 = 49 x = (−5 ± 7) / 6 x = 2/6 = ⅓ or x = −12/6 = −2 x = ⅓ or x = −2

Q4 MEDIUM Solve x² + 4x + 1 = 0, giving answers in exact form Show answer ▼Hide answer ▲

complete the square (or formula) (x + 2)² − 4 + 1 = 0 (x + 2)² = 3 x + 2 = ±√3 x = −2 ± √3

Q5 HARD Solve 2x² − 5x − 4 = 0, giving answers in exact form Show answer ▼Hide answer ▲

formula — a=2, b=−5, c=−4 b² − 4ac = 25 + 32 = 57 57 isn’t a perfect square → surds x = (5 ± √57) / 4 x = (5 ± √57) / 4 discriminant not a perfect square = factoring won’t work, formula is your friend!

⚠ Common mistakes

Forgetting one solution. A quadratic gives two values of x — write both, even if they look similar.
Sign errors in the formula. If b is negative, then −b becomes positive. Use parentheses: x = (−(−7) ± …) / 4.
Decimal answers when exact form is asked. Surds like √7 must stay as surds — never convert to 2.65.
Skipping the “= 0” step. The equation must equal zero before you solve. x² + 3x = 10 → first rewrite as x² + 3x − 10 = 0.
Ignoring negative discriminant. If b² − 4ac < 0, there are no real solutions — write that, don’t keep going.
Halving the wrong number when completing the square. Half the coefficient of x, not x² or the constant.

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Want the theory?

Read the full Quadratic Functions notes for the discriminant, the link to graph turning points, and the “why does completing the square work” explanation.

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