MathsICSE Class 10 · Quadratic Equations in One Variable

Solving Quadratic Equations by the Quadratic Formula (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac))/(2a). Write the equation in standard form, read a, b and c with their signs, find the discriminant, take its square root (from tables if asked), and round only the final answers to the accuracy asked.

Keep √D to four figures and work out both roots in full; round only the final answers (3.19 or 0.314).

Illustration drawn for QuestionBuddy.

Understand Solving Quadratics by the Quadratic Formula

Every quadratic equation $a x^{2} + b x + c = 0$ ( $a \neq = 0$ ) can be solved by the quadratic formula:

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a} .$

It comes from completing the square, so it works even when the equation does not factorise. In the ICSE paper it is the method expected whenever the question says 'give your answer correct to three significant figures' or 'correct to two decimal places' or 'use mathematical tables': those words tell you the roots are irrational. Both 2024 Q6(i) and 2025 Q2(i) were of exactly this type.

Step by step.

1. Bring the equation to the standard form $a x^{2} + b x + c = 0$ (all terms on one side, zero on the other, like terms collected).
2. Read $a$ , $b$ , $c$ with their signs. For $2 x^{2} - 7 x + 2 = 0$ : $a = 2$ , $b = - 7$ , $c = 2$ .
3. Find the discriminant $D = b^{2} - 4 a c$ separately. Here $D = 49 - 16 = 33$ .
4. Find $D$ to at least four significant figures: $33 = 5.745$ .
5. Substitute: $x = \frac{7 \pm 5.745}{4}$ . Work out the two roots separately: $\frac{12.745}{4} = 3.186$ and $\frac{1.255}{4} = 0.3138$ .
6. Round at the end: $x = 3.19$ or $x = 0.314$ (3 s.f.).

Significant figures. Counting starts at the first non-zero digit: $0.3138$ to 3 s.f. is $0.314$ ; $4.436$ is $4.44$ ; $0.05672$ is $0.0567$ . If you round $D$ too early (say to 7.55 or 7.5), the last figure of the answer may change. The examiners reported for 2024 Q6(i) candidates who took $60$ as $7.75$ instead of $7.746$ , and candidates who did not round to three significant figures.

Equations that need tidying first. Fractional equations such as $\frac{1}{x} + \frac{1}{x - 2} = 3$ become quadratics after multiplying by the LCM. State the values $x$ cannot take (here $0$ and $2$ ) and check that your roots are not among them.

Surd answers. If accuracy is not asked, leave the roots exactly, as in $x = 3 \pm 2$ . When the discriminant is a perfect square, the roots are rational and the equation would also factorise.

Solving Quadratics by the Quadratic Formula: formulas to remember

Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$; For $a x^{2} + b x + c = 0$ , $a \neq = 0$ .
Discriminant: $D = b^{2} - 4 a c$; Compute it on its own line.
Two roots: $α = \frac{- b + D}{2 a}, β = \frac{- b - D}{2 a}$; Work them out separately.
Sum and product of roots (check): $α + β = - \frac{b}{a}, α β = \frac{c}{a}$; A quick check on decimals.
Rounding rule: $D to 4 s.f. \to roots \to round last$; Never round before the final step.

How to solve Solving Quadratics by the Quadratic Formula questions

A fixed routine avoids sign and rounding errors.

1Rearrange to $a x^{2} + b x + c = 0$ ; clear fractions and brackets first.
2Write $a = \dots$ , $b = \dots$ , $c = \dots$ with signs.
3Compute $D = b^{2} - 4 a c$ ; remember $(- b)^{2}$ is positive.
4Find $D$ to four significant figures (tables or a reliable method).
5Write $x = \frac{- b \pm D}{2 a}$ with numbers, then find each root separately.
6Round each root to the required accuracy only at the end.
7Check with sum $= - \frac{b}{a}$ and, for fractional equations, that no root makes a denominator zero.

Solving Quadratics by the Quadratic Formula in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Forming and solving a quadratic with the formula

Suppose a 6 m ladder leans against a wall so that the top is 2 m higher up the wall than the foot is from the wall.

How far is the foot of the ladder from the wall, correct to 3 significant figures?

Try it first, then show the solutionSolution

Let the distance be $x$ m: $x^{2} + (x + 2)^{2} = 36$ .
$2 x^{2} + 4 x - 32 = 0 \Rightarrow x^{2} + 2 x - 16 = 0$ .
$x = \frac{- 2 \pm 4 + 64}{2} = - 1 \pm 17$ ; $17 = 4.123$ .
Distance must be positive: $x = 3.123 \approx 3.12$ .

Answer: About 3.12 m.

Memory hook: No neat factors? The formula always works.

Rejecting a negative root in a measurement

Imagine a rectangular poster whose length is 6 cm more than its width and whose area is 200 cm².

Find the width and length correct to 3 significant figures.

Try it first, then show the solutionSolution

$w (w + 6) = 200 \Rightarrow w^{2} + 6 w - 200 = 0$ .
$w = \frac{- 6 \pm 36 + 800}{2} = - 3 \pm 209$ ; $209 = 14.457$ .
$w = 11.457 \approx 11.5$ cm; length $= 17.457 \approx 17.5$ cm.

Answer: Width about 11.5 cm, length about 17.5 cm.

Memory hook: Lengths are never negative: keep the plus sign.

Tricks to remember Solving Quadratics by the Quadratic Formula

Short memory aids. Say each one aloud once, then use it on a question the same day.

Minus b, plus or minus, root, over 2a
Say it as a chant: 'minus b, plus or minus root of b-squared minus four-a-c, all over two a'.
Use it when: Writing the formula correctly every time.
Brackets round negative b
Write $(- 5)^{2} = 25$ with brackets so the square is never negative.
Use it when: Avoiding the 2024 Q1(iii) error of taking $(- q)^{2}$ as $- q^{2}$ .
Four in, three out
Take the square root to four figures; give answers to three.
Use it when: Questions asking for 3 significant figures.
'Use tables' means 'use the formula'
If the question mentions tables or significant figures, the equation will not factorise.
Use it when: Choosing the method quickly.

Solving Quadratics by the Quadratic Formula: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Roots in surd form · 1 mark · MCQBoard level

The roots of $x^{2} - 6 x + 7 = 0$ are:
(a) $3 \pm 2$
(b) $- 3 \pm 2$
(c) $6 \pm 8$
(d) $3 \pm 8$

Try it first, then show the solutionSolution

$D = 36 - 28 = 8$ ; $8 = 22$ .
$x = \frac{6 \pm 2 2}{2} = 3 \pm 2$ .

Answer: (a)

Example 2: Three significant figures · 3 marks · Short answerBoard level

Solve $3 x^{2} - 4 x - 2 = 0$ , giving your answers correct to three significant figures.

Try it first, then show the solutionSolution

$a = 3$ , $b = - 4$ , $c = - 2$ ; $D = 16 + 24 = 40$ ; $40 = 6.3246$ .
$x = \frac{4 \pm 6.3246}{6}$ .
$x = \frac{10.3246}{6} = 1.7208 \approx 1.72$ or $x = \frac{- 2.3246}{6} = - 0.38743 \approx - 0.387$ . (With $40 = 6.325$ the second root would land exactly on $- 0.3875$ ; one more figure settles the rounding.)

Answer: $x = 1.72$ or $x = - 0.387$

Example 3: Two decimal places · 3 marks · Short answerHarder board level

Solve $5 x^{2} + 2 x - 4 = 0$ , giving your answers correct to two decimal places.

Try it first, then show the solutionSolution

$D = 4 + 80 = 84$ ; $84 = 9.165$ .
$x = \frac{- 2 \pm 9.165}{10}$ .
$x = 0.7165 \approx 0.72$ or $x = - 1.1165 \approx - 1.12$ .

Answer: $x = 0.72$ or $x = - 1.12$

Example 4: A fractional equation · 4 marks · Short answerHarder board level

Solve $\frac{1}{x} + \frac{1}{x - 2} = 3$ , $x \neq = 0, 2$ , correct to two decimal places.

Try it first, then show the solutionSolution

Multiply by $x (x - 2)$ : $(x - 2) + x = 3 x (x - 2)$ .
$2 x - 2 = 3 x^{2} - 6 x \Rightarrow 3 x^{2} - 8 x + 2 = 0$ .
$D = 64 - 24 = 40$ ; $x = \frac{8 \pm 6.325}{6}$ .
$x = 2.3874 \approx 2.39$ or $x = 0.2792 \approx 0.28$ ; neither is 0 or 2.

Answer: $x = 2.39$ or $x = 0.28$

Example 5: A discriminant that is a perfect square in surds · 4 marks · Short answerHOTS / competency

Solve $x^{2} - (3 + 1) x + 3 = 0$ using the quadratic formula.

Try it first, then show the solutionSolution

$D = (3 + 1)^{2} - 43 = 3 + 23 + 1 - 43 = 4 - 23$ .
$4 - 23 = (3)^{2} - 23 + 1 = (3 - 1)^{2}$ , so $D = 3 - 1$ .
$x = \frac{( 3 + 1 ) \pm ( 3 - 1 )}{2}$ , so $x = 3$ or $x = 1$ .
Check: sum $= 3 + 1$ , product $= 3$ ✓.

Answer: $x = 3$ or $x = 1$

Example 6: Case study: fencing a garden · 4 marks · Case-basedHOTS / competency

A rectangular garden is 4 m longer than it is wide, and its area is 50 m².
(a) Form a quadratic equation in the width $w$ .
(b) Solve it, giving $w$ correct to 3 significant figures.
(c) Find the length of fencing needed to go round the garden, to the nearest metre.

Try it first, then show the solutionSolution

(a) $w (w + 4) = 50 \Rightarrow w^{2} + 4 w - 50 = 0$ .
(b) $w = \frac{- 4 \pm 16 + 200}{2} = \frac{- 4 \pm 216}{2}$ ; $216 = 14.70$ ; $w = 5.348 \approx 5.35$ m (reject the negative root).
(c) Length $= 9.348$ m; perimeter $= 2 (5.348 + 9.348) = 29.39 \approx 29$ m.

Answer: (a) $w^{2} + 4 w - 50 = 0$ (b) $w \approx 5.35$ m (c) about 29 m

Where students lose marks in Solving Quadratics by the Quadratic Formula

The slips examiners see most often, and what to write instead.

Mistake: Rounding the square root too early
An early rounding error can change the third significant figure.
Wrong working
$60 = 7.75$ , so $x = \frac{10 - 7.75}{4} = 0.5625 \approx 0.563$ .
Right working
$60 = 7.746$ : $x = 0.5635 \approx 0.564$ .
Do this: Keep four figures in $D$ ; round only the final answers.
Usually costs: Examiners named this for 2024 Q6(i).
Mistake: Sign errors in −b
If $b$ is negative, $- b$ is positive.
Wrong working
$2 x^{2} - 5 x - 4 = 0$ : $x = \frac{- 5 \pm 57}{4}$ .
Right working
$x = \frac{5 \pm 57}{4}$ .
Do this: Write $a$ , $b$ , $c$ with signs on a separate line.
Usually costs: Reported for 2025 Q2(i).
Mistake: Rounding to the wrong number of figures
'3 significant figures' is not '3 decimal places'.
Wrong working
$3.1375$ to 3 s.f. written as $3.138$ (that is 4 s.f.).
Right working
$3.1375 \approx 3.14$ (3 s.f.).
Do this: Count figures from the first non-zero digit.
Usually costs: The final answer mark.
Mistake: Forgetting to rearrange to standard form
The formula needs $a x^{2} + b x + c = 0$ .
Wrong working
$3 x^{2} = 8 x - 2$ : $a = 3$ , $b = 8$ , $c = - 2$ .
Right working
$3 x^{2} - 8 x + 2 = 0$ : $a = 3$ , $b = - 8$ , $c = 2$ .
Do this: Always bring everything to one side first.
Usually costs: All the marks.

Exam tips

2024 Q6(i) ( $2 x^{2} - 10 x + 5 = 0$ ) and 2025 Q2(i) ( $2 x^{2} - 5 x - 4 = 0$ ) both asked for 3 significant figures using tables; the examiners' expected answers were 4.44 and 0.564 (2024), and 3.14 and −0.638 (2025, worked from the table value √57 = 7.55).
Write the formula before substituting; it carries a method mark.
Show $D$ as a separate value (for example $57 = 7.550$ ).
Give both roots unless the context rules one out; then say why you rejected it.

Practice questions on Solving Quadratics by the Quadratic Formula

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
For $2 x^{2} - 3 x - 1 = 0$ , the value of $b^{2} - 4 a c$ is:
(a) 1
(b) 17
(c) −1
(d) 9
Hint
$b = - 3$ , $c = - 1$ .
Answer
(b) $9 + 8 = 17$
Q2 · 3 marks · Short answerBoard level
Solve $x^{2} - 4 x + 1 = 0$ , correct to three significant figures.
Hint
$12 = 3.464$ .
Answer
$x = 2 \pm 3$ : $x = 3.73$ or $x = 0.268$ .
Q3 · 3 marks · Short answerHarder board level
Solve $4 x^{2} - 3 x - 5 = 0$ , correct to two decimal places.
Hint
$D = 89$ .
Answer
$x = \frac{3 \pm 9.434}{8}$ : $x = 1.55$ or $x = - 0.80$ .
Q4 · 3 marks · Short answerHarder board level
Solve $x - \frac{1}{x} = 3$ , $x \neq = 0$ , correct to three significant figures.
Hint
Multiply by $x$ .
Answer
$x^{2} - 3 x - 1 = 0$ ; $x = \frac{3 \pm 13}{2}$ : $x = 3.30$ or $x = - 0.303$ .
Q5 · 1 mark · MCQHarder board level
The roots of $x^{2} + 2 x - 1 = 0$ are:
(a) $1 \pm 2$
(b) $- 1 \pm 2$
(c) $- 2 \pm 2$
(d) $- 1 \pm 22$
Hint
$D = 8$ .
Answer
(b)
Q6 · 4 marks · Short answerHOTS / competency
Solve $\frac{2}{x + 1} + \frac{3}{x + 2} = 1$ , $x \neq = - 1, - 2$ , correct to three significant figures.
Hint
Multiply by $(x + 1) (x + 2)$ .
Answer
$2 (x + 2) + 3 (x + 1) = x^{2} + 3 x + 2 \Rightarrow x^{2} - 2 x - 5 = 0$ ; $x = 1 \pm 6$ : $x = 3.45$ or $x = - 1.45$ .
Q7 · 4 marks · Short answerHarder board level
A ball is thrown upwards and its height after $t$ seconds is $h = 20 t - 5 t^{2}$ metres. At what times is it 12 m high? Give answers correct to 3 significant figures.
Hint
$5 t^{2} - 20 t + 12 = 0$ .
Answer
$t = \frac{20 \pm 160}{10}$ ; $160 = 12.65$ : $t = 0.735$ s (going up) and $t = 3.26$ s (coming down).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Write the quadratic formula.Show answer
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ .
2.For $x^{2} - 2 x - 2 = 0$ , find $D$ .Show answer
$4 + 8 = 12$ .
3.Roots of $x^{2} - 2 x - 2 = 0$ in surd form?Show answer
$1 \pm 3$ .
4.Round 0.04567 to 3 significant figures.Show answer
0.0457.
5.What is $(- 7)^{2} - 4 (1) (10)$ ?Show answer
$49 - 40 = 9$ .

Solving Quadratics by the Quadratic Formula: frequently asked questions

When should I use the quadratic formula instead of factorisation?

Use it when the equation does not factorise easily, and always when the question asks for an answer correct to a number of significant figures or decimal places, or says to use tables.

How many figures should I keep in the square root?

At least four significant figures, so that rounding the final answers to three significant figures is accurate.

What does 'correct to three significant figures' mean?

Keep three digits counting from the first non-zero digit, rounding the third. For example 4.436 becomes 4.44 and 0.5635 becomes 0.564.