S3 Maths | Solving Quadratic Equations: completing the square, formula and discriminant

S3 Maths | Solving Quadratic Equations: completing the square, formula and discriminant – tutor prep guide

2026-11-14

Tutor lesson-prep guides · Secondary Mathematics

Students memorise the formula, then cannot say why the plus-minus sign is there, so one root goes missing every time. This lesson builds the formula by hand from completing the square, uses the discriminant to settle the number of real roots with one number, and makes substituting back a habit.

Objectives — Students can: (1) complete the square to turn x squared plus bx plus c equals zero into a squared bracket equation and read off both roots; (2) copy a, b and c correctly and apply the formula, including negative and fractional roots; (3) use the discriminant D = b squared minus 4ac to state the number of real roots and link it to where the parabola meets the x-axis.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Factorising	How does x squared minus 5x plus 6 factorise?	(x minus 2)(x minus 3), two numbers adding to minus 5 and multiplying to 6
Square roots	How many square roots does 9 have?	Two, 3 and minus 3, so taking a root needs plus-minus
Area and perimeter	A rectangle is 6 by 4. What are its area and perimeter?	Area 24, perimeter 20

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Two warm-up equations	Write x squared minus 5x plus 6 equals zero and x squared equals 9, ask only how many solutions each has	State the solutions and count them
5-15 min	Completing the square	Turn x squared plus 6x plus 5 equals zero into (x plus 3) squared equals 4, adding the same number to both sides	Complete x squared plus 8x plus 7 equals zero independently
15-27 min	The formula	Derive x equals minus b plus-minus root of b squared minus 4ac over 2a, then run 2x squared minus 3x minus 2 equals zero	Apply the formula to one item after naming a, b and c
27-37 min	The discriminant	Show D values 25, 0 and minus 3 for two roots, a repeated root and no real roots, with three matching parabolas	Decide how many real roots each of three equations has
37-43 min	Roots and modelling	Give sum equals minus b over a and product equals c over a, then model a rectangle with perimeter 20 and area 24	Form and solve the equation from the word problem
43-45 min	Exit check	Collect three items: one completing the square, one discriminant, one sum and product	Answer and hand in

03 Teaching script (read it out)

Scene 1: Completing the square: add to both sides, do not memorise

The whole method is one sentence: adding (b/2) squared to x squared plus bx gives a perfect square, because (x plus b/2) squared equals x squared plus bx plus (b/2) squared. Train the habit of adding the same number to both sides. For x squared plus 6x plus 5 equals zero: b is 6, b/2 is 3, add 9 to both sides, giving (x plus 3) squared equals 4, so x plus 3 equals plus or minus 2 and x equals minus 1 or minus 5. Three checkpoints: keep the sign with b/2 since its square stays positive; write plus-minus when taking the root or one solution disappears; and divide through first when the leading coefficient is not 1, as in 2x squared plus 8x minus 10 equals zero.

Have students do x squared plus 8x plus 7 equals zero and state that 16 was added to both sides, giving minus 1 and minus 7

Scene 2: The formula: a mis-copied coefficient loses the whole item

For ax squared plus bx plus c equals zero with a not zero, x equals minus b plus-minus the root of b squared minus 4ac, all over 2a. The first move is not substitution but rewriting the equation with zero on the right, then copying a, b and c one at a time with their signs. With 2x squared minus 3x minus 2 equals zero: a is 2, b is minus 3, c is minus 2, the discriminant is 9 plus 16 equals 25, its root is 5, so x equals 3 plus or minus 5 over 4, giving 2 and minus 0.5. Three common copying errors: reading coefficients before rearranging, writing b instead of minus b, and writing a instead of 2a.

Ask for the a, b and c of x squared minus 4x minus 12 equals zero, then the discriminant 64 and the roots 6 and minus 2

Scene 3: The discriminant: one number, three stories

D equals b squared minus 4ac settles the number of real roots because it sits inside the root sign: D greater than zero gives two distinct real roots and two crossings of the x-axis; D equals zero gives one repeated root where the vertex touches the axis; D less than zero gives no real roots with the parabola entirely above or below the axis. For example x squared plus 4x plus 4 has D equal to zero and the repeated root minus 2, while x squared plus x plus 1 has D equal to minus 3 and no real roots. Say clearly that no real roots is not the same as no solution, since complex roots exist but S3 works in the reals.

Give three equations and require only the discriminant before stating the number of real roots, with a quick sketch each time

04 Question types and worked steps

Type	Example	Steps	Answer
Completing the square	Solve x squared plus 6x plus 5 equals zero by completing the square.	(1) Take b/2 equals 3 (2) Add 9 to both sides (3) Take the root with plus-minus (4) Write both roots	x equals minus 1 or minus 5
The formula	Solve 2x squared minus 3x minus 2 equals zero.	(1) Check standard form (2) Copy a 2, b minus 3, c minus 2 (3) Discriminant 25 (4) Apply the formula	x equals 2 or minus 0.5
Discriminant decision	How many real roots does x squared plus x plus 1 equals zero have?	(1) Copy a, b, c (2) Discriminant minus 3 (3) It is below zero (4) No real roots	No real roots, D equals minus 3
Word problem modelling	A rectangle has perimeter 20 and area 24. Find its sides.	(1) Let width be w and length w plus 2 (2) Write w(w plus 2) equals 24 (3) Rearrange to w squared plus 2w minus 24 equals zero (4) Solve for 4 and 6	Width 4, length 6

Sanity check (teach this): Three checks after every item. Substitution first: put the roots back into the original equation and both sides must match, since 2 and minus 0.5 both satisfy 2x squared minus 3x minus 2 equals zero. Root count second: a quadratic has at most two real roots, so three answers or one and a half mean an arithmetic slip, and the sign of D must agree with the number given. Context third: a negative root that stands for a length must be rejected explicitly with a reason rather than reported as a side of minus 8.

05 Independent practice (3 questions)

Question	Answer	What to watch
Solve x squared plus 8x plus 7 equals zero by completing the square.	x equals minus 1 or minus 7, after adding 16	Whether 16 was added to both sides and plus-minus written
How many real roots does 3x squared minus 6x plus 3 equals zero have?	One, since D equals 36 minus 36 equals zero, root x equals 1	Whether D was computed and the repeated root named
A rectangle has perimeter 20 and area 24. Find the width.	Width 4, with length 6	Whether the negative root was rejected with a reason

06 Common mistakes and what to say

Mistake	Student writes	Say this
Dropping the plus-minus	x squared equals 9, so x equals 3.	A square has two roots, so x is plus or minus 3. Dropping one solves only half the equation, and both must be written.
Copying coefficients before rearranging	x squared equals 5x minus 6, so b is 5 and c is minus 6.	Move everything left first to get x squared minus 5x plus 6 equals zero, then b is minus 5 and c is 6. Signs come from the move, not the original layout.
Reading the discriminant as a root	D equals minus 3, so x equals minus 3.	The discriminant only counts real roots; it is not a root. A negative D means no real roots, not a root of minus 3.
Keeping a negative length	The sides came out as minus 8 and 3, so the side is minus 8.	A length cannot be negative. Compute both roots, then judge each against the context and state why the impossible one is rejected.

07 Homework (paper, section and question numbers)

Factorising quadratics (Q1-Q5) — Factorising first saves time later, so drill it before the formula
Equations of straight lines (Q2-Q4) — Sign handling for gradients matches copying a, b and c carefully
Secondary Mathematics index (pick two algebra papers, timed) — Twenty minutes timed; check plus-minus and substitution every time

08 Exit check (last 5 minutes)

Question	Answer	Decision
What are the two roots of x squared equals 16?	4 and minus 4	One root only means the plus-minus needs reteaching
What is the discriminant of x squared plus 2x plus 5 equals zero, and how many real roots are there?	D equals minus 16, so no real roots	A negative value stated with no real roots is a pass
What is the maximum number of real roots a quadratic can have?	Two	Three answers mean the formula or the root step is wrong

09 Tutor FAQ

Why teach completing the square when the formula is faster?

Because the formula comes from it. Students who build the square themselves stop writing a instead of 2a or b instead of minus b. Some questions also demand the method by name, so skipping it forfeits the item.

Does a negative discriminant mean no solution?

Strictly it means no real roots; solutions still exist among the complex numbers. At S3 the safe answer is no real roots, and the scope can be stated plainly.

Should a negative root be written in a word problem?

Write both roots from the working, then judge each against the context. Discarding it silently costs method marks; rejecting it with a reason earns them.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.

Practice papers　Paper 1 (basics)　·　Paper 2 (intermediate)　·　Paper 3 (challenge)　·　Paper 4 (error clinic)