IB Maths AA SL
Paper 1 & 2
15 min read
Solving Quadratic Equations
A quadratic equation looks like ax2 + bx + c = 0. There are three reliable methods for solving it on a non-calculator paper — factorising, the quadratic formula, and completing the square — plus the GDC shortcut on Paper 2. The trick is picking the right method for each question.
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What you need to know
- A quadratic equation must be in the form ax2 + bx + c = 0 before you can solve it
- Factorising works when the quadratic factors with integers — fastest when it works
- The quadratic formula always works — it’s in the formula booklet
- Completing the square always works too — best when factoring fails or when the question asks for it
- On calculator papers, the GDC solves any quadratic instantly
- If b2 − 4ac < 0, the equation has no real solutions
Which Method Should I Use?
Pick the fastest tool for the job:
When: Calculator paper
Fastest. Just type in a, b, c and read the roots. Use this on Paper 2.
When: Roots are nice
If you can factorise with integers, this is the fastest non-calculator method.
When: Default fallback
Always works. Especially useful when factorising fails or roots are irrational.
When: Asked, or for vertex info
Always works too. Best if the question asks, or if it gives vertex/min info too.
Method 1: Solving by Factorising
If you can factorise the quadratic into the form a(x − p)(x − q) = 0, then by the zero product rule, at least one of the factors must equal zero:
If (x − p)(x − q) = 0, then x = p or x = q
- Get the equation into the form ax2 + bx + c = 0 (right side equals 0).
- Factorise the left side.
- Set each factor equal to zero and solve.
Reminder of the sign flip: if the factor is (x − 3), the root is x = +3 (not −3). The root is the value that makes the bracket zero.
Method 2: The Quadratic Formula
The quadratic formula always works, no matter how ugly the numbers are. It’s given in the formula booklet:
Anatomy of the Quadratic Formula
x = −b ± √(b2 − 4ac)2a
−b (red)
Negative of the x coefficient. Watch the sign carefully if b is itself negative.
± (purple)
Gives you two solutions: one with +, one with −.
b2 − 4ac (green)
The discriminant. If it’s negative, no real solutions. If 0, one repeated solution.
2a (blue)
Twice the x2 coefficient. Sits as the denominator.
- Rearrange to ax2 + bx + c = 0.
- Identify a, b, c — including signs!
- Calculate the discriminant first: b2 − 4ac. This avoids sign mistakes.
- Substitute into the formula and simplify.
Examiner trick: when the values are awkward, work out the discriminant b2 − 4ac on its own first. It cuts down on negative-sign errors and makes the rest easier.
Method 3: Solving by Completing the Square
Once the quadratic is in vertex form a(x − h)2 + k = 0, you can solve it by isolating the squared bracket and taking square roots:
- Complete the square on the left side.
- Isolate the squared bracket: (x − h)2 = −ka.
- If the right side is negative, there are no real solutions.
- Otherwise, take ± square root of both sides.
- Solve for x: x = h ± √(−k/a).
Don’t forget the ±! Taking the square root of both sides gives two answers — a positive and a negative. Forgetting the minus loses you half the solutions.
Worked Examples
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Example 1 — Solve by factorising (monic)
Solve x2 − 5x + 6 = 0.
Answer:
Step 1: factorise. Find p, q with p+q = −5, p×q = 6.
p = −2, q = −3
(x − 2)(x − 3) = 0
Step 2: set each factor to zero.
x − 2 = 0 → x = 2
x − 3 = 0 → x = 3
x = 2 or x = 3
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Example 2 — Solve by factorising (non-monic)
Solve 4x2 + 4x − 15 = 0.
Answer:
Step 1: factorise (covered in Factorising Quadratics).
(2x + 5)(2x − 3) = 0
Step 2: set each factor to zero.
2x + 5 = 0 → 2x = −5 → x = −5/2
2x − 3 = 0 → 2x = 3 → x = 3/2
x = −5/2 or x = 3/2
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Example 3 — Quadratic formula (rearrange first)
Solve 7 − 3x − 5x2 = 0.
Answer:
Step 1: identify a, b, c carefully.
a = −5, b = −3, c = 7
Step 2: discriminant first.
b² − 4ac = (−3)² − 4(−5)(7)
= 9 + 140 = 149
Step 3: substitute into the formula.
x = (−(−3) ± √149) / (2 × −5)
x = (3 ± √149) / (−10)
x = (3 ± √149) / (−10)
Or equivalently: x = −(3 ± √149) / 10. Both forms are accepted.
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Example 4 — Quadratic formula (clean numbers)
Solve 2x2 − 7x + 3 = 0 using the quadratic formula.
Answer:
Step 1: identify a, b, c.
a = 2, b = −7, c = 3
Step 2: discriminant.
b² − 4ac = 49 − 24 = 25
Step 3: substitute.
x = (7 ± √25) / 4
x = (7 ± 5) / 4
Step 4: split into the two solutions.
x = (7 + 5) / 4 = 12/4 = 3
x = (7 − 5) / 4 = 2/4 = 1/2
x = 3 or x = 1/2
Check: this could have been factorised as (2x − 1)(x − 3) = 0.
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Example 5 — Solve by completing the square
Solve 3x2 + 12x − 5 = 0 by completing the square.
Answer:
Step 1: complete the square (covered in Completing the Square).
3(x + 2)² − 17 = 0
Step 2: isolate the squared bracket.
3(x + 2)² = 17
(x + 2)² = 17/3
Step 3: take ± square root.
x + 2 = ± √(17/3)
Step 4: solve for x.
x = −2 ± √(17/3)
Don’t forget the ± — that gives both solutions.
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Tips
- Try factorising first — if it works in 5 seconds, great. If not, move to the formula.
- Compute the discriminant alone first when using the quadratic formula. It’s easier to spot sign errors when the calculation is small.
- Always make right side equal to zero. 2x2 = x + 6 must become 2x2 − x − 6 = 0 before you can solve.
- Check your answer by plugging it back into the original equation. Quick and catches careless mistakes.
- If the discriminant is negative, stop and write “no real solutions” — don’t waste time trying to compute square roots of negatives.
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Common mistakes
- Sign errors with negative b. If b = −3, then −b = +3, not −3. The minus inside the formula flips the sign of b.
- Forgetting the ±. Both the quadratic formula and completing-the-square need both solutions. Skipping one loses half the marks.
- Forgetting to set the equation to zero first. The quadratic formula only works on ax2 + bx + c = 0 form.
- Reading the sign of a, b, c wrong. In −5x2 + 3x − 7, a = −5, b = 3, c = −7. Carry the negative signs over.
- Confusing factor with root. Factor (x − 3) means root x = +3. The sign flips.
- Stopping at (x − h)2 = number. You still need to take the square root and add h to finish.
Final word: Three methods that always work, and one shortcut (the GDC). Pick the easiest for the question you’ve got — and watch the signs like a hawk.
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