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Short Study Notes — Quadratic Equations
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Quadratic Equations — IB MYP Mathematics (Extended): factorisation, formula, and the discriminant
A quadratic equation has the form ax2+bx+c=0. This MYP Mathematics Extended note covers three solution methods (factorisation, completing the square, formula), the discriminant, and applications.
What you’ll learn
Mapped to the IB MYP Mathematics subject guide (2026 onwards).
MYP Mathematics A — Solve quadratics by factorisation.
MYP Mathematics A — Apply the quadratic formula.
MYP Mathematics A — Use the discriminant to classify roots.
MYP Mathematics D — Model real situations with quadratics.
Solving by factorisation
Easiest method when factorisation works.
If you can factorise the quadratic into a product of two brackets, you can solve it by the zero-product rule: if AB=0, then A=0 or B=0.
Worked example. Solve x2−5x+6=0.
Factorise: x2−5x+6=(x−2)(x−3).
Zero-product: x−2=0 or x−3=0.
So x=2 or x=3.
Worked example. Solve 2x2+7x+3=0.
Factorise (AC method): 2x2+7x+3=(2x+1)(x+3).
2x+1=0 → x=−21​. x+3=0 → x=−3.
So x=−21​ or x=−3.
Worked example (difference of squares). Solve x2−9=0.
x2−9=(x−3)(x+3)=0.
x=3 or x=−3.
(Alternatively: x2=9⇒x=±3.)
Always:
Move everything to one side (= 0).
Factorise.
Set each factor to 0.
Solve for x.
Move all terms to one side.
Factorise the quadratic.
Zero-product rule: each bracket = 0.
The quadratic formula
Works EVERY time.
When factorisation is hard or impossible, use the quadratic formula:
x=2a−b±b2−4ac​​.
It comes from completing the square on the general form ax2+bx+c=0.
Worked example. Solve 2x2+5x−3=0 using the formula.
a=2, b=5, c=−3.
Discriminant: b2−4ac=25−4(2)(−3)=25+24=49.
x=4−5±49​​=4−5±7​.
x=42​=0.5 or x=4−12​=−3.
Worked example. Solve x2−4x+1=0.
a=1, b=−4, c=1.
b2−4ac=16−4=12.
x=24±12​​=24±23​​=2±3​.
This is a case where factorisation would fail — the roots are irrational. Always have the formula in your toolkit.
Strategy: try factorisation first (faster); use the formula if it doesn't factorise nicely.
Formula: x=2a−b±b2−4ac​​.
Works for ANY quadratic.
Try factorisation first; formula as a backup.
The discriminant
Tells you how many roots without solving.
The expression under the square root in the formula is called the discriminant:
Δ=b2−4ac.
It controls the number of real roots:
Δ
Number of real roots
Geometric meaning
>0
2 distinct real roots
parabola crosses x-axis twice
=0
1 repeated root (or 'double root')
parabola touches x-axis once (tangent)
<0
0 real roots
parabola never crosses x-axis
The discriminant $\Delta = b^2 - 4ac$ controls how many times the parabola $y = ax^2 + bx + c$ crosses the x-axis.
Worked example. How many roots does x2+6x+9=0 have?
Δ=36−36=0 → ONE repeated root.
Factorise: (x+3)2=0⇒x=−3 (double root).
Worked example. Does x2+2x+5=0 have real roots?
Δ=4−20=−16<0 → NO real roots.
Many Extended-level questions ask you to use the discriminant to find unknown coefficients.
Δ=b2−4ac.
Positive →2 roots; zero →1 repeated; negative →0 real roots.
Tells you the number of x-axis crossings without solving.
How it’s examined
Criterion A: solve by factorisation or formula. Criterion C: clear step-by-step solving. Criterion D: model a real situation with a quadratic (projectile, area).
Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.
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Step-by-step worked examples — Quadratic Equations
Step-by-step solutions to past-paper-style questions on quadratic equations, written exactly the way a tutor would explain them at the board.
1Solve by factorisation
Building confidence• factorise
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Question
Solve x2+8x+15=0.
Step-by-step solution
Step 1
Find two numbers multiplying to 15 and adding to 8: 3+5=8, 3×5=15.
Step 2
Factorise: (x+3)(x+5)=0.
Step 3
Zero-product: x=−3 or x=−5.
Answer
x=−3 or x=−5.
2Use the quadratic formula
Building confidence• formula
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Question
Solve 3x2−7x+2=0.
Step-by-step solution
Step 1
a=3, b=−7, c=2.
Step 2
Δ=49−24=25, so Δ​=5.
Step 3
x=67±5​. So x=2 or x=31​.
Answer
x=2 or x=31​.
3Find k using the discriminant
Stretch• discriminant
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Question
Find the values of k for which x2+kx+9=0 has exactly one repeated root.
Step-by-step solution
Step 1
One repeated root means Δ=0.
Step 2
Δ=k2−36=0. So k2=36, k=±6.
Answer
k=6 or k=−6.
4Real context — projectile
Stretch• application
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Question
A ball thrown upwards has height h=−5t2+20t+1 metres after t seconds. When does it hit the ground?
t=1020±20.5​. Two roots: ≈4.05 s or −0.05 s.
Step 5
Reject negative time. Ball hits ground at t≈4.05s.
Answer
t≈4.05 s.
Key Definitions and Keywords — Quadratic Equations
Definitions to memorise and the exact keywords mark schemes credit for quadratic equations answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.
Quadratic equation
Examiner keyword
An equation of the form ax2+bx+c=0 with aî€ =0.
Root (of an equation)
Examiner keyword
A value of x that satisfies the equation. For a quadratic, also called a solution or zero.
Discriminant Δ
Examiner keyword
Δ=b2−4ac. Determines the number of real roots.
Completing the square
Rewriting ax2+bx+c as a(x−h)2+k — useful for finding the vertex of a parabola.
Common Mistakes and Misconceptions — Quadratic Equations
The traps other students keep falling into on quadratic equations questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.
✕Trying to factorise x2+5x=14 as if it equals 0.
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Why it happens
Skipping the 'all on one side' step.
How to avoid it
Always move everything to one side first: x2+5x−14=0, then factorise.
✕Forgetting the minus sign in −b in the quadratic formula.
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Why it happens
Mental arithmetic shortcut.
How to avoid it
Write down each value of a,b,c FIRST, with their signs. Then substitute carefully into the formula.
✕Missing the negative root.
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Why it happens
Stopping after the first solution.
How to avoid it
A quadratic typically has TWO roots. The ± in the formula gives both. Always state BOTH (unless the discriminant is 0).
Practice questions
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