Chapter 4: Quadratic Equations
1. Quadratic Equation — Definition
A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Standard form: ax2 + bx + c = 0, a ≠ 0
Here a is the coefficient of x2, b is the coefficient of x, and c is the constant term. Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation.
Checking whether an equation is quadratic: Simplify both sides completely and bring every term to the left-hand side. The equation is quadratic only if the highest power of x remaining is exactly 2. For example, x(x + 1) + 8 = (x + 2)(x − 2) simplifies to x + 12 = 0, so it is not quadratic (the x2 terms cancel).
2. Roots of a Quadratic Equation
A real number α is called a root of the quadratic equation ax2 + bx + c = 0, a ≠ 0, if aα2 + bα + c = 0. We also say that x = α satisfies the equation.
- The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.
- A quadratic equation can have at most two real roots (two distinct, two equal, or none).
3. Solution by Factorisation
Stepwise method (splitting the middle term):
- Write the equation in standard form ax2 + bx + c = 0.
- Find two numbers p and q such that p + q = b and p × q = ac.
- Split the middle term: ax2 + px + qx + c = 0, and factorise by grouping into two linear factors.
- Equate each linear factor to zero and solve for x.
Worked example: 6x2 − x − 2 = 0. Here ac = −12 and b = −1; the numbers are −4 and 3. So 6x2 − 4x + 3x − 2 = 0 ⇒ 2x(3x − 2) + 1(3x − 2) = 0 ⇒ (3x − 2)(2x + 1) = 0. Hence x = 2/3 or x = −1/2.
Exam tip: Always verify a factorisation answer by substituting the roots back, or by checking sum of roots = −b/a and product of roots = c/a. This one-line check catches sign errors and costs no marks.
4. Solution by the Quadratic Formula
Quadratic formula: The roots of ax2 + bx + c = 0, a ≠ 0, are given by
x = [−b ± √(b2 − 4ac)] / 2a, provided b2 − 4ac ≥ 0.
Stepwise method: (1) Write the equation in standard form and identify a, b, c with correct signs. (2) Compute D = b2 − 4ac. (3) If D ≥ 0, substitute in the formula and simplify; if D < 0, write "no real roots".
Worked example: 2x2 + x − 4 = 0. Here a = 2, b = 1, c = −4, so D = 1 + 32 = 33 > 0. Roots: x = (−1 ± √33)/4.
5. Discriminant and Nature of Roots
The quantity D = b2 − 4ac is called the discriminant of the quadratic equation ax2 + bx + c = 0. It decides the nature of the roots.
| Value of D = b2 − 4ac | Nature of roots | The roots |
|---|---|---|
| D > 0 | Two distinct real roots | x = (−b + √D)/2a and x = (−b − √D)/2a |
| D = 0 | Two equal real roots (repeated/coincident root) | x = −b/2a, −b/2a |
| D < 0 | No real roots | — |
Exam tip: In "find k for equal roots" questions, put D = 0 and solve for k. If the coefficient of x2 itself contains k, reject any value of k that makes a = 0 (the equation would no longer be quadratic). Example: kx(x − 2) + 6 = 0 gives kx2 − 2kx + 6 = 0; D = 4k2 − 24k = 0 gives k = 0 or k = 6, and k = 0 is rejected, so k = 6.
6. Word Problems (Situational Problems) — Standard Method
Day-to-day situational problems that lead directly to a quadratic equation are in the syllabus. Stepwise method:
- Let the unknown quantity be x (choose the smaller/simpler quantity).
- Translate the given condition into an equation and reduce it to standard form ax2 + bx + c = 0.
- Solve by factorisation or by the quadratic formula.
- Check admissibility: reject roots that are negative lengths, negative speeds, non-integers where integers are required, etc., and write the rejection reason.
Common board question patterns
- Consecutive integers: e.g. sum of squares of two consecutive positive integers is 365 ⇒ x2 + (x + 1)2 = 365.
- Speed–distance–time: "had the speed been 5 km/h more, time taken would be 1 hour less" ⇒ 360/x − 360/(x + 5) = 1.
- Upstream/downstream (boat): speeds (u − v) upstream and (u + v) downstream, difference of times given.
- Ages: product or sum-of-ages conditions at different times.
- Area/perimeter of rectangles: l + b and l × b given ⇒ x(s − x) = A where s is the half-perimeter.
- Nature-of-roots reasoning: "Is it possible to design ... ?" — form the equation, compute D; possible only if D ≥ 0.
Exam tip: In every word problem, the final statement line ("Hence the speed of the train is 40 km/h") and the rejection line ("x = −45 is rejected since speed cannot be negative") each carry marks. Never skip them.
7. Points to Remember
- a ≠ 0 is essential; if a = 0 the equation becomes linear.
- √D is taken with both + and − signs, giving the two roots.
- If the roots of ax2 + bx + c = 0 are equal, then the repeated root is x = −b/2a.
- Equations like x2 + 4 = 0 have no real roots since D = −16 < 0.
- Irrational roots occur when D > 0 but D is not a perfect square (e.g. D = 33).
Deleted from the Rationalised Syllabus (do not study)
- Situational problems based on equations reducible to quadratic equations (e.g. equations in 1/x, x4, or requiring substitution to become quadratic) — deleted.
- Solution by completing the square as a formal method — removed from NCERT; the quadratic formula is now used directly.
- Retained: standard form, solution by factorisation, solution by the quadratic formula, discriminant and nature of roots, and direct situational/day-to-day problems on quadratic equations.