What this chapter is about
A quadratic equation is a polynomial equation of degree two. After learning linear equations in earlier classes, you now move to equations where the variable appears with a highest power of two. These equations arise naturally when dealing with areas, projectile motion, profit-loss calculations and many situations where the relationship between quantities is not simply proportional.
This chapter teaches you how to recognise a quadratic equation, express real-world problems in quadratic form, and solve these equations using different methods. You will learn factorisation, completing the square, and the quadratic formula. You will also study the discriminant, which tells you about the nature of roots without actually solving the equation.
By the end of this chapter, you should be able to form quadratic equations from word problems, solve them accurately, and interpret whether the roots are real, equal or imaginary based on the discriminant.
Key ideas
- A quadratic equation in variable x has the standard form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
- The values of x that satisfy the equation are called roots or solutions; a quadratic equation has at most two roots.
- Factorisation method works when ax² + bx + c can be written as a product of two linear factors.
- Completing the square converts any quadratic into the form (x + p)² = q, from which roots can be found by taking square roots.
- The quadratic formula gives roots directly: x = (−b ± √(b² − 4ac)) / 2a, and works for every quadratic equation.
- The discriminant D = b² − 4ac determines the nature of roots: if D > 0, two distinct real roots; if D = 0, two equal real roots; if D < 0, no real roots.
- A quadratic equation can have at most two real roots, never three or more.
Formulas and facts to remember
Standard form: ax² + bx + c = 0, where a ≠ 0. Meaning: The equation must have a non-zero coefficient for x².
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a Meaning: Substitute values of a, b, c to find both roots directly.
Discriminant: D = b² − 4ac Meaning: This single number tells you about the nature of roots.
Nature of roots based on D:
- D > 0 → two distinct real roots
- D = 0 → two equal real roots (also called repeated root)
- D < 0 → no real roots
Sum of roots: −b/a Product of roots: c/a Meaning: These relate coefficients to roots without solving.
Worked examples
Example 1: Solving by factorisation
Solve: x² − 7x + 12 = 0
Step 1: Find two numbers whose product is 12 and sum is −7. The numbers are −3 and −4 (since −3 × −4 = 12 and −3 + −4 = −7).
Step 2: Write the equation as (x − 3)(x − 4) = 0.
Step 3: Set each factor to zero. x − 3 = 0 gives x = 3 x − 4 = 0 gives x = 4
The roots are x = 3 and x = 4.
Example 2: Using the quadratic formula
A rectangular garden has length 5 metres more than its breadth. If the area is 84 square metres, find the dimensions.
Let breadth = x metres. Then length = (x + 5) metres. Area = x(x + 5) = 84 x² + 5x − 84 = 0
Here a = 1, b = 5, c = −84. D = b² − 4ac = 25 − 4(1)(−84) = 25 + 336 = 361 √D = 19
x = (−5 ± 19) / 2 x = (−5 + 19) / 2 = 7 or x = (−5 − 19) / 2 = −12
Since breadth cannot be negative, x = 7. Breadth = 7 metres, Length = 12 metres.
Example 3: Finding nature of roots
Without solving, determine the nature of roots of 2x² − 4x + 5 = 0.
Here a = 2, b = −4, c = 5. D = b² − 4ac = 16 − 4(2)(5) = 16 − 40 = −24
Since D < 0, the equation has no real roots.
Common mistakes
Writing ax² + bx + c = 0 with a = 0 and calling it quadratic → Remember, a must be non-zero; otherwise it becomes linear.
Forgetting the ± sign in the quadratic formula and finding only one root → Always compute both (−b + √D)/2a and (−b − √D)/2a.
Taking square root of a negative discriminant and writing a real answer → If D < 0, state that no real roots exist; do not proceed further.
Accepting negative values for length, age or count in word problems → Check whether the root makes sense in the given context and reject impossible values.
Errors in sign when calculating b² − 4ac, especially when b or c is negative → Write out each term carefully before combining.
Quick revision
- Standard form: ax² + bx + c = 0 with a ≠ 0.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.
- D > 0 means two different real roots; D = 0 means one repeated root; D < 0 means no real roots.
- Always check if roots are valid for the given word problem context.
- Factorisation is quick when factors are easy to spot; otherwise use the formula.