What is a quadratic equation? (standard form)
A quadratic equation is an equation where the biggest power of x is 2. "Quad" means square.
Its standard form is ax² + bx + c = 0. Here a, b and c are real numbers and a ≠ 0. If a were 0, the x² part would vanish and it would be a linear equation.
Always bring every term to one side first. Example: 2x² = 3x − 1 becomes 2x² − 3x + 1 = 0, so a = 2, b = −3, c = 1.
A number that makes the equation true is called a root (or solution). A quadratic has at most two roots.
How to check if an equation is quadratic
Open all brackets and simplify. If the highest power of x left is 2, it is quadratic. Careful: (x + 1)² = x² + 3 simplifies to 2x − 2 = 0, which is not quadratic, because the x² cancels.
Solving by factorisation (splitting the middle term)
For ax² + bx + c = 0, find two numbers that multiply to a × c and add to b. Break the middle term into these two, group, and take out common factors.
Example: x² + 5x + 6 = 0. We need two numbers with product 6 and sum 5: they are 2 and 3.
x² + 2x + 3x + 6 = 0 → x(x + 2) + 3(x + 2) = 0 → (x + 2)(x + 3) = 0.
If two numbers multiply to give 0, at least one of them is 0. So x + 2 = 0 or x + 3 = 0, which gives x = −2 or x = −3.
The 3D tiles show why this works: the pieces of x² + 5x + 6 fit exactly into a rectangle with sides (x + 2) and (x + 3).
Solving by the quadratic formula
Some equations do not factorise easily. The quadratic formula always works:
x = (−b ± √(b² − 4ac)) / 2a
Steps: (1) write a, b, c with their signs; (2) find D = b² − 4ac; (3) if D ≥ 0, find √D; (4) put the values in the formula once with + and once with −.
Where does the formula come from?
It comes from completing the square. Divide by a: x² + (b/a)x + c/a = 0. Add and subtract (b/2a)²: (x + b/2a)² = (b² − 4ac)/4a². Take the square root of both sides: x + b/2a = ±√(b² − 4ac)/2a. Move b/2a across and you get the formula. The ± appears because both a positive and a negative number give the same square.
Discriminant and nature of roots
The part under the root, D = b² − 4ac, is called the discriminant. "Discriminate" means "tell apart": D tells us what kind of roots we get without solving.
- D > 0: two different real roots. The curve cuts the x-axis at two points.
- D = 0: two equal real roots, x = −b/2a. The curve just touches the x-axis.
- D < 0: no real roots, because no real number squared is negative. The curve stays away from the x-axis.
A common board question: "Find k so that kx² + 6x + 1 = 0 has equal roots." Set D = 0: 36 − 4k = 0, so k = 9.
Word problems on quadratic equations
Four steps turn any story into a quadratic:
- Name the unknown: let it be x (say what x means and its unit).
- Write the other quantities using x.
- Form the equation from the condition, and bring it to ax² + bx + c = 0.
- Solve and check: throw away answers that make no sense (a negative length, age or speed).
Typical types: area of a rectangle, consecutive numbers, ages, speed–time (trains, boats in a stream), and work done by two people.
Try it: the tile puzzle at home
Cut 1 big square (side 5 cm), 5 strips (5 cm × 1 cm) and 6 small squares (1 cm) from paper. They stand for x², 5x and 6. Try to arrange all 12 pieces into one rectangle. Measure its sides: one side is x + 2, the other is x + 3. Now try x² + 7x + 12 (7 strips, 12 small squares). Which rectangle do you get?
In the 3D above, go to the last step and predict first: with a = 1, b = 4, c = 4, how many times will the curve meet the x-axis? Then move the sliders and check.
What is asked in the board exam
This chapter is part of the Algebra unit (about 20 marks). Expect: checking if an equation is quadratic (1 mark), finding k for equal roots or nature of roots (2 marks), solving by factorisation or formula (2–3 marks), and one word problem (3–5 marks, often speed or area).
Key formulas and definitions
- Standard form: ax² + bx + c = 0, a ≠ 0
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
- Discriminant: D = b² − 4ac
- D > 0: two distinct real roots; D = 0: two equal roots x = −b/2a; D < 0: no real roots
- Splitting the middle term: find p, q with p × q = a × c and p + q = b
Worked examples
1. Is (x − 2)² + 1 = 2x − 3 a quadratic equation?
Open the bracket: x² − 4x + 4 + 1 = 2x − 3. Bring all terms left: x² − 6x + 8 = 0. The highest power is 2 and a = 1 ≠ 0, so yes, it is quadratic.
2. Solve x² − 7x + 12 = 0 by factorisation.
Need two numbers with product 12 and sum −7: −3 and −4. x² − 3x − 4x + 12 = 0 → x(x − 3) − 4(x − 3) = 0 → (x − 3)(x − 4) = 0. So x = 3 or x = 4.
3. Solve 2x² + x − 6 = 0 by factorisation.
a × c = 2 × (−6) = −12. Two numbers with product −12 and sum 1: 4 and −3. 2x² + 4x − 3x − 6 = 0 → 2x(x + 2) − 3(x + 2) = 0 → (x + 2)(2x − 3) = 0. So x = −2 or x = 3/2.
4. Solve x² − 2x − 3 = 0 using the quadratic formula.
a = 1, b = −2, c = −3. D = (−2)² − 4(1)(−3) = 4 + 12 = 16, √D = 4. x = (2 ± 4)/2. So x = 6/2 = 3 or x = −2/2 = −1.
5. Find the nature of roots of 3x² − 4x + 2 = 0.
D = (−4)² − 4(3)(2) = 16 − 24 = −8. D < 0, so the equation has no real roots.
6. Find k so that x² − kx + 9 = 0 has two equal roots.
For equal roots D = 0: (−k)² − 4(1)(9) = 0 → k² = 36 → k = 6 or k = −6.
7. The length of a rectangular garden is 4 m more than its width. Its area is 96 m². Find its sides.
Let width = x m, length = x + 4 m. x(x + 4) = 96 → x² + 4x − 96 = 0 → (x + 12)(x − 8) = 0 → x = 8 or x = −12. Width cannot be negative, so width = 8 m and length = 12 m.
8. A train covers 360 km at a steady speed. If its speed were 10 km/h more, it would take 3 hours less. Find its speed.
Let speed = x km/h. Time = 360/x. Then 360/x − 360/(x + 10) = 3. Multiply by x(x + 10): 360(x + 10) − 360x = 3x(x + 10) → 3600 = 3x² + 30x → x² + 10x − 1200 = 0 → (x + 40)(x − 30) = 0. Speed cannot be negative, so the speed is 30 km/h.
Common mistakes
- Forgetting to bring every term to one side before reading a, b, c. In x² = 5x − 6, b is −5 and c is 6, not 5 and −6.
- Dropping the sign of b in the formula. If b = −3, then −b = +3 and b² = 9 (not −9).
- Dividing both sides by x. This throws away the root x = 0. Factorise x out instead.
- Keeping impossible answers in word problems, like a negative length or age. Always check and reject them.