Sec 4 G3 Maths Bukit Timah | Quadratic Formula

Secondary 4 G3 Mathematics Tutor Bukit Timah — solving quadratics becomes easier when you connect factorisation, the formula and completed-square form rather than treating them as unrelated tricks.

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Did you know? One quadratic can tell three different stories
The formula, written so nothing disappears
The discriminant predicts how many real roots to expect
Completing the square reveals the turning point
Which method should a student choose under exam conditions?
A word problem where one algebraic root is not valid
Common mistakes that survive even after plenty of worksheets
A short seven-day revision loop
Parent questions, answered quickly

Did you know? One quadratic can tell three different stories

Many Secondary 4 students learn factorisation, the quadratic formula and completing the square as separate drills. Did you know they are actually three ways of understanding the same expression? Factorisation shows where a graph crosses the horizontal axis, completing the square reveals its turning point, and the quadratic formula can find real roots even when factors are not obvious.

The quick answer for a Secondary 4 G3 Mathematics learner in Bukit Timah is this: first put the equation in the form ax² + bx + c = 0, with a not equal to zero. Choose a valid solving method, preserve every sign, and check each result in the original equation. If the question describes a physical quantity, interpret the roots before reporting them.

The 2027 Singapore–Cambridge SEC G3 Mathematics K310 syllabus explicitly includes factorisation, the quadratic formula, completing the square for a monic quadratic of the given form, and graphical methods. A 2026 Secondary 4 O-Level candidate should instead use the applicable Mathematics 4052 requirements.

The formula, written so nothing disappears

For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac))/(2a). The whole numerator is divided by 2a; the ± symbol means there may be two values. Write the three coefficients on their own line before entering anything into a calculator.

For example, in 2x² − 5x − 3 = 0, the coefficients are a = 2, b = −5 and c = −3. The quantity under the square root is (−5)² − 4(2)(−3) = 25 + 24 = 49. Then x = (5 ± 7)/4, giving x = 3 or x = −1/2.

Check both answers. At x = 3, 2(9) − 15 − 3 = 0. At x = −1/2, 2(1/4) + 5/2 − 3 = 0. The equation also factorises as (2x + 1)(x − 3) = 0, so the independent methods agree.

The discriminant predicts how many real roots to expect

The discriminant is D = b² − 4ac. If D is positive, the quadratic has two different real roots. If D is zero, it has one repeated real root. If D is negative, it has no real roots. In a real-number examination question, do not try to take a real square root of a negative discriminant.

For x² − 4x + 4 = 0, D = 16 − 16 = 0. The equation is (x − 2)² = 0, so x = 2 is repeated. Its graph just touches the x-axis.

For x² + 2x + 5 = 0, D = 4 − 20 = −16. Rewriting it as (x + 1)² + 4 = 0 makes the reason visible: a square plus 4 cannot equal zero for real x. State no real roots rather than inventing two intersections.

Completing the square reveals the turning point

Consider y = x² − 6x + 5. Half the coefficient of x is −3. Since (x − 3)² = x² − 6x + 9, we have y = (x − 3)² − 4. That exposes a minimum of −4 when x = 3, so the turning point is (3, −4).

To solve x² − 6x + 5 = 0, use (x − 3)² = 4. This means x − 3 = 2 or x − 3 = −2, so x = 5 or x = 1. Do not forget the negative branch after taking a square root.

Check the factorised form y = (x − 1)(x − 5). The two roots, 1 and 5, are equally far from the symmetry line x = 3. Different algebraic forms give the same geometry.

Which method should a student choose under exam conditions?

Factorisation is efficient when the factors are recognisable, as in x² − 7x + 12 = 0, whose roots are 3 and 4. The formula is useful when factorisation is awkward, for example x² − 2x − 1 = 0, giving x = 1 ± √2.

Completing the square is particularly revealing when a question asks for a minimum or maximum. The graphical method connects roots to x-intercepts. No single method is best for every question; the learner should recognise the shape of the task before committing to a sequence of steps.

A good tutor asks why a method was chosen. A correct answer obtained from a long, error-prone route may reveal that method selection still needs practice.

A word problem where one algebraic root is not valid

A rectangle has area 40 cm², and its length is 3 cm greater than its width. Let its positive width be x cm. Then the length is x + 3, so x(x + 3) = 40.

Rearrange: x² + 3x − 40 = 0, or (x + 8)(x − 5) = 0. The algebraic roots are −8 and 5. The width cannot be −8 cm, so x = 5 cm. The length is 8 cm. Check: 5 × 8 = 40 cm².

Did you know? Rejecting a root is not the same as forgetting it. Show that the algebra produced two values, then explain the physical constraint that excludes one. That makes the reasoning auditable.

Common mistakes that survive even after plenty of worksheets

One mistake is changing b = −5 to b = 5 when entering the formula. Another is dividing only the square-root term by 2a. A third is applying ± only to part of an expression. A fourth is reporting a turning point as a single x-value rather than a coordinate pair.

Another frequent error is stopping at a factorisation without solving for x. The expression (x − 2)(x − 7) = 0 implies x = 2 or x = 7; those are the requested roots. If the equation came from a word problem, check units and whether each root is allowed.

Write a one-line mistake diagnosis such as ‘I forgot the negative square-root branch’ or ‘I used the wrong sign of b.’ Replace it with a rule and test that rule on a changed quadratic two days later.

A short seven-day revision loop

Day 1: solve four easily factorisable quadratics. Day 2: substitute signed coefficients into the quadratic formula. Day 3: classify discriminants before solving. Day 4: complete the square on expressions of the form x² + px + q.

Day 5: connect roots, symmetry and the turning point using sketches. Day 6: solve a rectangle or area word problem requiring contextual interpretation. Day 7: attempt unfamiliar mixed questions without a method printed in the heading.

The purpose is not to complete seven piles of worksheets. It is to see whether a student can explain the method and solve a new question independently after the original worked example has been closed.

Parent questions, answered quickly

Is a repeated root one answer or two? It is one distinct real solution, occurring twice algebraically. Must every quadratic have real roots? No: a negative discriminant means there are no real roots.

Can students check the formula by factorising? Yes, when factorisation is possible. Should they round surds immediately? Keep exact roots when requested; otherwise follow the question’s accuracy instructions.

Does completing the square only help with graphs? No. It also helps solve equations and interpret minima or maxima. Will better algebra guarantee an A1? No. It provides a measurable route to fewer repeat mistakes, but no tutor can guarantee a grade.

How three-pupil Mathematics tutorials work in Bukit Timah

The unchanged eduKateSG Mathematics small-group reference describes weekly 1.5-hour classes limited to three pupils, guided correction and independent home practice. The stated teaching location is 8 Fourth Avenue, Singapore 268674, near Sixth Avenue MRT. These details are not a claim of a second venue or an available seat.

Ask the learner to bring their latest actual error in a quadratic problem. The tutor can identify whether the first mistake came from coefficients, sign changes, factorisation, a missing ± branch or contextual interpretation, then retest on a different problem. Enquire about a parent–student consultation.

Official sources and the next reading

All worked examples are original learning exercises, not official or predicted examination questions. The student should always verify their actual examination-year rules.