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1. Introduction

A quadratic equation in the variable x is an equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. The name comes from the Latin word "quadratus", meaning square, because the equation contains the square of the variable. This chapter develops systematic methods to solve such equations and to analyse the nature of their solutions.

The chapter begins with the method of factorisation, which expresses the quadratic as a product of two linear factors and equates each factor to zero. It then presents the method of completing the square, which prepares the groundwork for deriving the famous quadratic formula. The quadratic formula gives the roots of any quadratic equation in terms of its coefficients and is the most general and reliable method of solution.

Beyond solving, the chapter introduces the discriminant D = b^2 - 4ac, which determines the nature of the roots without actually computing them. Quadratic equations find wide application in physics, engineering, and economics, and in this chapter they are used to solve word problems involving areas, distances, numbers, and time-speed situations.

2. Standard Form and Roots

A quadratic equation in standard form is written as ax^2 + bx + c = 0, with a not equal to zero. A solution or root of the equation is a value of x that satisfies the equation, i.e., the value for which the left side becomes zero.

A quadratic equation has at most two real roots. This is consistent with the fundamental theorem of algebra and the fact that the graph of a quadratic is a parabola, which can cross the x-axis at most twice.

Example: The equation 2x^2 - 5x + 3 = 0 has a = 2, b = -5, c = 3. Substituting x = 1 gives 2 - 5 + 3 = 0, so 1 is a root. Substituting x = 3/2 gives 2(9/4) - 5(3/2) + 3 = 9/2 - 15/2 + 3 = 0, so 3/2 is the other root.

3. Solving by Factorisation

Steps

  1. Write the equation in standard form ax^2 + bx + c = 0.
  2. Factorise the left side into two linear factors using middle-term splitting.
  3. Set each factor equal to zero, since a product is zero only if at least one factor is zero.
  4. Solve the two linear equations to obtain the roots.

Example: Solve x^2 - 5x + 6 = 0. We need two numbers whose product is 6 and sum is -5: they are -2 and -3. x^2 - 5x + 6 = (x - 2)(x - 3) = 0 Hence x = 2 or x = 3.

Middle-Term Splitting

For ax^2 + bx + c, find two numbers p and q with p + q = b and p x q = a x c. Then split bx as px + qx and factor by grouping.

4. Solving by Completing the Square

This method transforms the equation into the form (x + m)^2 = n, from which the roots can be extracted by taking square roots.

Steps

  1. Write the equation as ax^2 + bx = -c.
  2. Divide throughout by a so that the coefficient of x^2 is 1.
  3. Add (b/2a)^2 to both sides to complete the square on the left.
  4. Take the square root of both sides and solve for x.

Example: Solve x^2 + 4x - 5 = 0. x^2 + 4x = 5 x^2 + 4x + 4 = 5 + 4 (x + 2)^2 = 9 x + 2 = 3 or x + 2 = -3 x = 1 or x = -5.

The quantity added, (b/2a)^2, is always half the coefficient of x, squared.

5. The Quadratic Formula

By completing the square on the general equation ax^2 + bx + c = 0, we derive the quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)]/2a

Steps to Use the Formula

  1. Write the equation in standard form and identify a, b, c.
  2. Substitute into the formula.
  3. Simplify, taking care to handle both the plus and minus cases.

Example: Solve 2x^2 - 5x + 3 = 0 using the formula. a = 2, b = -5, c = 3. x = [5 +/- sqrt(25 - 24)]/4 = [5 +/- 1]/4 x = 6/4 = 3/2 or x = 4/4 = 1.

The quadratic formula works for every quadratic equation, whether factorable or not, and it is the method that always succeeds.

6. Nature of Roots: The Discriminant

The expression D = b^2 - 4ac is called the discriminant of the quadratic equation ax^2 + bx + c = 0.

Example: For x^2 - 5x + 6 = 0, D = 25 - 24 = 1 > 0, so two distinct real roots. For x^2 + 4x + 4 = 0, D = 16 - 16 = 0, so equal roots x = -2, -2. For x^2 + x + 1 = 0, D = 1 - 4 = -3 < 0, so no real roots.

The discriminant is a powerful tool because it answers questions about the number of roots instantly without solving the equation.

7. Word Problems

Quadratic word problems follow a consistent approach:

  1. Read the problem and identify the unknown, assigning it the variable x.
  2. Frame the quadratic equation from the given conditions.
  3. Solve the equation using factorisation or the formula.
  4. Reject solutions that do not make sense in the context (e.g., negative lengths or distances).
  5. Write the final answer with appropriate units.

Common problems involve finding unknown numbers whose product and sum are given, the dimensions of a rectangle given area and perimeter, time taken for journeys, and the price of articles when the total amount is given.

Quick Revision Tables

Table 1: Solving Methods Comparison

Method Idea Best Used When
Factorisation Split middle term, set factors to zero Roots are simple integers/fractions
Completing the Square Form (x + m)^2 = n Asked explicitly or a is 1 and b is even
Quadratic Formula x = [-b +/- sqrt(b^2 - 4ac)]/2a Any equation; always works

Table 2: Nature of Roots Using Discriminant

Discriminant D = b^2 - 4ac Nature of Roots Example
D > 0 Two distinct real roots x^2 - 5x + 6 = 0, D = 1
D = 0 Two equal real roots x^2 + 4x + 4 = 0, D = 0
D < 0 No real roots x^2 + x + 1 = 0, D = -3

Mind Map

graph TD A["Quadratic Equations ax^2 + bx + c = 0"] --> B["Methods of Solution"] A --> C["Discriminant D = b^2 - 4ac"] A --> D["Word Problems"] B --> B1["Factorisation"] B --> B2["Completing the Square"] B --> B3["Quadratic Formula"] C --> C1["D > 0: two distinct real roots"] C --> C2["D = 0: equal real roots"] C --> C3["D < 0: no real roots"] D --> D1["Frame equation from conditions"] D --> D2["Reject invalid solutions"]

Important Diagrams (SVG)

Diagram 1: Quadratic Formula Derivation Flow

Deriving the Quadratic Formula by Completing the Square Start: ax^2 + bx + c = 0 x^2 + (b/a)x = -c/a Add (b/2a)^2 to both sides (x + b/2a)^2 = (b^2 - 4ac)/4a^2 x + b/2a = +/- sqrt(b^2 - 4ac)/2a x = [-b +/- sqrt(b^2 - 4ac)]/2a Golden Rule: The formula x = [-b +/- sqrt(b^2 - 4ac)]/2a works for every quadratic equation.

Diagram 2: Nature of Roots Visual

Nature of Roots from the Discriminant D > 0 Two distinct real roots D = 0 Two equal real roots D < 0 No real roots Golden Rule: D > 0 gives two roots, D = 0 gives equal roots, D < 0 gives no real roots.

Common Mistakes

  1. Forgetting that a cannot be zero in ax^2 + bx + c = 0; if a = 0, the equation is linear, not quadratic.
  2. In factorisation, finding numbers whose product is c instead of a x c when a is not 1, leading to a wrong splitting.
  3. In completing the square, adding (b/2a)^2 to one side only, breaking the equality of the equation.
  4. Forgetting the +/- sign when taking the square root of both sides, losing one root.
  5. Taking the square root of a negative number as a real number; if the discriminant is negative, state clearly that there are no real roots.
  6. In word problems, keeping a negative root that makes no sense (like a negative length) in the final answer instead of rejecting it.
  7. Substituting values into the quadratic formula with wrong signs, e.g., writing -b as -(-5) incorrectly or mishandling 4ac when c is negative.

Exam Tips

  1. Always rewrite the equation in standard form ax^2 + bx + c = 0 and list a, b, c before applying the formula.
  2. For the discriminant questions, mention D = b^2 - 4ac and then the conclusion; examiners award step marks.
  3. Prefer factorisation for simple equations as it is faster, but switch to the formula if splitting is hard.
  4. In word problems, clearly reject invalid roots with a one-line reason before writing the answer.
  5. Practise questions like "find the value of k for which the equation has equal roots" by setting D = 0.
  6. Remember: a quadratic has equal roots when D = 0, so substitute D = 0 and solve for the unknown coefficient.
  7. Verify each root by substituting it back into the original equation; this catches most careless errors.

Conclusion

Quadratic equations are a cornerstone of algebra, and this chapter provides every tool needed to solve them completely. From factorisation to completing the square to the universal quadratic formula, each method reinforces an understanding of how the coefficients of a quadratic determine its roots. The discriminant gives an elegant way to predict the nature of the roots, and word problems connect the algebra to practical situations such as areas, speeds, and numbers. Mastering this chapter ensures readiness for the applications that follow in trigonometry and geometry, and it is a consistent source of high-scoring questions in board examinations.