Did you know? A quadratic equation may have two different answers—and both can be mathematically correct. In a real question, however, one answer may describe an impossible negative length. Solving the equation is only the first job; interpreting the answers is the second.
This guide is for families researching a Secondary 3 G1 Mathematics tutor in Bukit Timah. Under Full Subject-Based Banding, G1 is a subject level. The SEAB 2027 G1 syllabus directory and official K110 Mathematics syllabus set out the whole course, while individual secondary schools decide their teaching sequence and current weighted-assessment topics. Use the school’s actual worksheets and assessment scope to choose what to practise now.
Quick answer: what does the quadratic formula do?
A quadratic equation has an x² term and can be written in the form ax² + bx + c = 0, with a non-zero a. The quadratic formula gives its solutions: x = (−b ± √(b² − 4ac))/(2a). The symbol ± means we calculate once with addition and once with subtraction. The expression beneath the square root is b² − 4ac.
The K110 G1 whole-course syllabus specifies solving quadratic equations by using the formula. Students are also expected to formulate quadratic equations for problems. The aim is accurate identification of a, b and c, controlled substitution, and sensible checking—not a race to memorise complicated patterns.
Worked example: a rectangular display
Question. The length of a fictional rectangular display is 3 metres more than its width. The area is 10 m². Find its dimensions.
- Let the width be x metres. Then the length is (x + 3) metres.
- Write the area equation: x(x + 3) = 10.
- Put it into standard quadratic form: x² + 3x − 10 = 0.
- Read the coefficients: a = 1, b = 3 and c = −10.
- Substitute into the formula: x = (−3 ± √(3² − 4 × 1 × (−10)))/(2 × 1).
- Calculate the square root: 3² − 4(1)(−10) = 9 + 40 = 49, so √49 = 7.
- Find both roots: x = (−3 + 7)/2 = 2, or x = (−3 − 7)/2 = −5.
For the equation alone, x = 2 and x = −5 both work. For this rectangle, however, width cannot be negative. The only meaningful width is 2 m, making the length 5 m. Check: 2 × 5 = 10 m².
Why two values appear
A quadratic equation describes a curved relationship. Imagine the graph y = x² + 3x − 10. Where it crosses the x-axis, y = 0, and the x-values are −5 and 2. This links equation-solving to the earlier guide on quadratic graphs, but the practical rectangle problem still requires us to reject the negative width.
A student who supplies just ‘x = 2’ may have interpreted the context correctly, but it is useful to show both mathematical solutions before stating why one is unsuitable. That reasoning makes the final answer stronger.
Check both roots directly
- At x = 2: x² + 3x − 10 = 4 + 6 − 10 = 0.
- At x = −5: x² + 3x − 10 = 25 − 15 − 10 = 0.
- Conclusion: both numbers solve the equation, but only x = 2 describes a positive rectangle width.
The three sign errors that do the most damage
Treating a negative c as positive
In our example, c is −10, so −4ac = −4(1)(−10) = +40. Write the negative coefficient in brackets every time. Without those brackets, students can turn 49 into an incorrect negative or small number.
Forgetting that 2a is one denominator
The formula divides the entire numerator by 2a. Input (−b + √(…))/(2a) as a grouped expression. Dividing only the square-root term changes the solution.
Calculating only the plus case
The ± symbol usually requires two evaluations when both are real and distinct. A student who chooses the positive sign every time may accidentally miss a valid solution or overlook a restriction in a word problem.
An easy second example
Solve 2x² − 5x − 3 = 0. Here a = 2, b = −5 and c = −3. The formula gives x = (5 ± √(25 + 24))/4 = (5 ± 7)/4. The solutions are x = 3 and x = −0.5. Check x = 3: 18 − 15 − 3 = 0. Check x = −0.5: 0.5 + 2.5 − 3 = 0.
Notice that a negative b becomes positive when calculating −b. This is why copying the equation into a small coefficient table before using the formula is such an effective habit.
What the number beneath the square root tells us
The expression b² − 4ac is often called the discriminant. When it is positive, the formula produces two distinct real roots; when it is zero, the two expressions coincide; when it is negative, there are no real roots. In this G1 guide, the main purpose is to use the formula correctly and interpret ordinary real solutions, not to introduce advanced proof.
A useful five-question self-check
- Did I first rearrange the equation so the right side is zero?
- Did I copy a, b and c with their signs?
- Did I put negative coefficients in brackets?
- Did I calculate both plus and minus cases, when appropriate?
- Did I check the answers in the original equation and read the wording of the question?
Questions parents often ask
Is the quadratic formula required at G1?
Yes. The official 2027 K110 syllabus explicitly lists solving quadratic equations in one variable using the formula, which is also included in the examination formulae list.
Should we start by factorising everything?
Factorisation is useful background, but this assessment skill specifically includes the formula. Build the ability to identify coefficients and substitute correctly even when simple factorisation is not obvious.
What if my child’s two calculator answers look strange?
Check the sign of b and c, then the numerator and denominator brackets. Finally substitute each value back into the original equation; the left side should be zero or very close to it after rounding.
What to do next in Bukit Timah
Bring a recent school question with every line of working to the tutor. Ask the learner to explain the first decision, not only the final answer. eduKateSG’s immutable small-group Mathematics tutorial reference describes a Secondary 1 learning approach near Sixth Avenue MRT, including close attention and guided corrections. It does not confirm the current availability of any Secondary 3 G1 class. Parents can contact eduKate Singapore to enquire about a suitable arrangement.
Continue reading
- Changing the Subject of a Formula
- Similar Triangles and Scale Factors
- Cumulative Frequency and Quartiles
- Quadratic graphs and turning points
For national subject information, refer directly to the SEAB 2027 K110 syllabus.