Chapter 4 — Quadratic Equations

A quadratic equation is a polynomial equation of degree 2 in a variable x. Its standard form is ax² + bx + c = 0 where a, b and c are real numbers and a ≠ 0. The graph of the quadratic function y = ax² + bx + c is a parabola.

Methods to solve quadratic equations (Class 10 level):

• Factorisation: Write the quadratic as product of two linear factors, e.g. (px + q)(rx + s) = 0 and use zero-product property.
• Completing the square: Rewrite in the form a(x + h)² = k and solve for x.
• Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a), used always and useful when factorisation is not obvious.

Discriminant Δ = b² − 4ac determines the nature of roots:

• Δ > 0: Two distinct real roots (if Δ is a perfect square, rational).
• Δ = 0: One repeated (real) root.
• Δ < 0: Two complex (non-real) roots.

Important properties: If α and β are roots of ax² + bx + c = 0, then sum of roots α + β = −b/a and product αβ = c/a. The vertex of the parabola y = ax² + bx + c is at x = −b/(2a) and y value f(−b/(2a)); the axis of symmetry is x = −b/(2a). If a > 0 the parabola opens upward (vertex is minimum); if a < 0 it opens downward (vertex is maximum).

Real-life examples:

• Projectile motion: Height vs time of a thrown object (ignoring air resistance) is a quadratic function; the object rises to a maximum (vertex) then falls.
• Area problems: If one side of a rectangular field is expressed in terms of the other variable, the area leads to a quadratic equation when set to a fixed value.
• Profit/revenue modelling: Some revenue or profit functions approximate a quadratic in terms of price or quantity; maximizing profit corresponds to the parabola's vertex.

Definition. A quadratic equation in one variable x has the standard form ax² + bx + c = 0 where a ≠ 0 and a, b, c are real numbers. The graph of y = ax² + bx + c is a parabola.

Main methods to solve.

• 1. Factorisation: Express the quadratic as a product of two linear factors: (px + q)(rx + s) = 0. Then use the zero-product property: if AB = 0 then A = 0 or B = 0. Best when factors are integers or simple rationals.
• 2. Completing the square: Rewrite ax² + bx + c into a perfect square plus constant, typically by dividing by a (if a ≠ 1), then adding and subtracting the square term. This gives (x + d)² = e, then solve x = −d ± √e. Useful for deriving vertex form and when factorisation is not obvious.
• 3. Quadratic formula: For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). This always works and is especially handy when other methods are hard.
• 4. Square-root method: If the equation can be written as (linear expression)² = k, take square roots: linear expression = ±√k. Works when b = 0 in standard form after rearrangement or after completing the square.
• 5. Graphical method: Solve by plotting y = ax² + bx + c and finding x-intercepts (where y = 0). Useful for visual understanding and approximate roots.

Discriminant and nature of roots. The discriminant Δ = b² − 4ac determines the nature of roots:

• Δ > 0: two distinct real roots
• Δ = 0: one real (repeated) root
• Δ < 0: two complex conjugate roots (no real roots)

When to choose which method? Factorisation is fastest if simple factors exist. Use completing the square to find vertex or when coefficients allow it. Use the quadratic formula when factorisation is hard or to guarantee exact answers. Use the square-root method for equations reducible to a square. Use graphing for visual/approximate solutions.

Derivation highlight (quadratic formula via completing square). Divide by a: x² + (b/a)x + c/a = 0. Move c/a: x² + (b/a)x = −c/a. Add (b/2a)² to both sides to make a perfect square: (x + b/2a)² = (b² − 4ac)/4a². Take square root and solve to get x = (−b ± √(b² − 4ac)) / (2a).

What it is: The factorisation method solves a quadratic equation by writing the quadratic polynomial as a product of two linear factors and then using the zero-product property: if AB = 0 then A = 0 or B = 0.

Standard form: ax² + bx + c = 0 (a ≠ 0). To use factorisation we try to express ax² + bx + c as (px + q)(rx + s).

Steps (simple case a = 1):

• Write x² + bx + c. Find two numbers m and n such that m + n = b and m·n = c.
• Then x² + bx + c = (x + m)(x + n). Set each factor = 0 to get roots x = −m, x = −n.

General case (a ≠ 1) — AC or split-middle method:

1. Compute product a·c.
2. Find two integers p and q with p + q = b and p·q = a·c.
3. Rewrite bx as px + qx, then factor by grouping into two binomials.
4. Extract common factors to get (px + q)(rx + s) and solve each linear factor = 0.

Special factorisation patterns to recognise:

• Difference of squares: a² − b² = (a − b)(a + b)
• Perfect square trinomial: a² ± 2ab + b² = (a ± b)²
• Common factor: take out greatest common divisor first (e.g., 2x²+4x = 2x(x+2))

When factorisation is convenient: If the quadratic has real rational roots (often when the discriminant Δ = b² − 4ac is a perfect square), factorisation over integers or rationals is usually possible and quick.

What to do if it does not factor nicely: Use the quadratic formula x = [−b ± √(b² − 4ac)]/(2a) or complete the square. Factorisation may still be possible over real irrational or complex numbers, but not always needed in Class 10 problems.

What it is: Completing the square is a method to rewrite a quadratic expression ax² + bx + c in the form a(x − h)² + k (called the vertex form). This makes it easy to find the vertex, axis of symmetry and to solve quadratic equations.

Why it works (idea): For x² + bx, adding (b/2)² makes the expression a perfect square: x² + bx + (b/2)² = (x + b/2)². For ax² + bx + c, first factor out a (if a ≠ 0) from the x² and x terms, then complete the square inside the bracket.

Procedure (a = 1):

1. Take x² + bx + c.
2. Add and subtract (b/2)²: x² + bx + (b/2)² − (b/2)² + c.
3. Group the perfect square: (x + b/2)² − (b/2)² + c, so the vertex form is (x + b/2)² + (c − (b/2)²).

Procedure (general a ≠ 1):

1. Start with ax² + bx + c. Factor a from the first two terms: a[x² + (b/a)x] + c.
2. Inside the bracket add and subtract (b/2a)²: a[x² + (b/a)x + (b/2a)² − (b/2a)²] + c.
3. Rewrite: a[(x + b/2a)² − (b/2a)²] + c = a(x + b/2a)² − a(b/2a)² + c.
4. Simplify constant: a(x + b/2a)² + (c − b²/4a).

Vertex and axis of symmetry: If ax² + bx + c = a(x − h)² + k, then the vertex is (h, k) with h = −b/(2a). The axis of symmetry is x = h.

Connection to quadratic formula: Completing the square applied to ax² + bx + c = 0 gives x = [−b ± sqrt(b² − 4ac)]/(2a).

When to use: To solve quadratic equations (especially when factoring is hard), to find vertex/maximum or minimum values, and to convert to vertex form for graphing.

What it is: The quadratic formula gives the solutions (roots) of any quadratic equation in standard form ax^2 + bx + c = 0, where a ≠ 0. It is

x = (-b ± √(b^2 - 4ac)) / (2a).

Derivation (brief): Start from ax^2 + bx + c = 0. Divide by a, complete the square for x^2 + (b/a)x, then solve for x to obtain the formula above. The expression under the square root, Δ = b^2 - 4ac, is called the discriminant and determines the nature of the roots.

• Discriminant Δ = b^2 - 4ac: if Δ > 0 → two distinct real roots; Δ = 0 → one real repeated root; Δ < 0 → two complex conjugate roots.
• Useful related formulas: axis of symmetry x = -b/(2a); vertex coordinates ( -b/(2a), (4ac - b^2)/(4a) ); sum and product of roots: α + β = -b/a, αβ = c/a.
• When to use: Use the quadratic formula when factoring is hard or impossible by inspection, or to get exact roots including irrational or complex ones.

Tips for using the formula: identify a, b, c carefully (including signs); compute Δ first to know root type; simplify √Δ before final division when possible.

Discriminant (D)
The discriminant of a quadratic equation ax^2 + bx + c = 0 (a ≠ 0) is defined as D = b^2 − 4ac. It is a number that determines the nature (type and number) of the roots of the quadratic.

Nature of roots by value of D

• D > 0: Two distinct real roots. If D is a perfect square (and a, b, c are rational/integer), the roots are rational; if D is not a perfect square the roots are irrational.
• D = 0: Two equal (repeated) real roots; the quadratic has a double root (both roots equal to −b/(2a)). The graph of the parabola is tangent to the x-axis at the vertex.
• D < 0: Two complex conjugate (non-real) roots. The parabola does not meet the x-axis.

Why it works (brief)
Using the quadratic formula x = [−b ± sqrt(b^2 − 4ac)]/(2a), the expression under the square root is the discriminant. Its sign determines whether the square root is real and whether the ± gives two distinct values or a repeated value.

Connections with graph of y = ax^2 + bx + c
The discriminant is directly related to intersection points of the parabola with the x-axis:
- D > 0: parabola cuts x-axis at two points.
- D = 0: parabola touches (is tangent to) x-axis at one point (vertex).
- D < 0: parabola does not cut the x-axis.

Useful related facts
- Axis of symmetry (x-coordinate of vertex): x = −b/(2a).
- y-coordinate of vertex: y_v = −D/(4a). (So sign of D affects whether vertex y-value is positive/negative relative to a.)
- Sum of roots α + β = −b/a; product αβ = c/a.

Procedure to determine nature of roots
1. Identify a, b, c in ax^2 + bx + c = 0.
2. Compute D = b^2 − 4ac.
3. Use the sign and value of D to classify the roots (and to check if they are rational or irrational when D > 0).

What the topic states
For a quadratic equation ax^2 + bx + c = 0 (a ≠ 0), if its roots are α and β, then the sum and product of the roots are related to the coefficients by Vieta's relations:

α + β = -b/a, αβ = c/a.

Derivation (brief)
Divide the equation by a: x^2 + (b/a)x + c/a = 0. If the roots are α and β, the quadratic factors as (x − α)(x − β) = x^2 − (α + β)x + αβ. Comparing coefficients gives α + β = −b/a and αβ = c/a.

Consequences and useful facts

• Axis of symmetry (vertical line through vertex): x = (α + β)/2 = −b/(2a). The vertex x-coordinate is the average of the roots.
• Vertex y-coordinate: f(−b/(2a)) = −Δ/(4a), where Δ = b^2 − 4ac is the discriminant.
• Nature of roots from discriminant Δ: Δ > 0 ⇒ two distinct real roots, Δ = 0 ⇒ equal (repeated) root α = β = −b/(2a), Δ < 0 ⇒ two complex conjugate roots. Vieta's relations still hold for complex roots (sum and product real if coefficients are real).
• Special cases: if c = 0 then α = 0 and β = −b/a (one root is zero). If a = 1 (monic), then α + β = −b and αβ = c.
• To form a quadratic with given roots r1 and r2: k[x^2 − (r1 + r2)x + r1 r2] = 0 for any nonzero constant k (usually k = 1).

Interpretation
The sum of the roots locates the axis of symmetry and the product (scaled by a) gives the constant term c; thus coefficients encode where the parabola crosses the x-axis and the y-intercept: f(0) = c = aαβ.

A quadratic equation in one variable x has the general form ax² + bx + c = 0 (a ≠ 0). If its roots (zeros) are α and β, then the equation can be written as k(x − α)(x − β) = 0 for some nonzero constant k. The most common (monic) choice is k = 1, giving x² − (α + β)x + αβ = 0.

Key ideas and methods:

• From given roots α and β: directly use k(x − α)(x − β) = 0. Expand to standard form and choose k (often 1) to get integer coefficients if required.
• From given sum S and product P of roots: let α + β = S and αβ = P. The monic equation is x² − Sx + P = 0.
• From a relation between roots (e.g., roots are in a given ratio, one root is a multiple of the other, roots differ by d, one root is reciprocal of the other, etc.): express α and β in terms of a parameter (say k), compute sum and product, then form the equation.
• If one root is complex (nonreal) and coefficients are real, the conjugate is also a root. Example: if 2 + i is a root, the other is 2 − i, and the quadratic is (x − (2 + i))(x − (2 − i)) = x² − 4x + 5 = 0.

Practical tips:

• If roots are fractions, choose k to clear denominators and get integer coefficients.
• If only one root r is given and you are told, for example, the sum S of roots, then the other root is S − r; use this to find product and form the equation.
• For roots in ratio m:n, let roots be mk and nk. Then sum = (m + n)k and product = mn k². Use these to find k from any additional condition (like given sum) and then form the equation.

Connections to graph of a quadratic:

• For y = a(x − α)(x − β): x-intercepts are α and β. The axis of symmetry (vertical) is x = (α + β)/2. The vertex x-coordinate is (α + β)/2 and vertex y-coordinate equals −Δ/(4a) where Δ is the discriminant (b² − 4ac). For this factor form, Δ = a²(α − β)², so y_vertex = −a(α − β)²/4.
• If roots are real and distinct the parabola crosses the x-axis twice; if roots are equal it is tangent to the x-axis (repeated root); if roots are complex it does not meet the x-axis.

What this topic means
Application problems leading to quadratic equations are word problems from real life that, after assigning a variable and translating relations into algebra, reduce to an equation of the form ax^2 + bx + c = 0. Solving that quadratic gives the required numerical answers (lengths, ages, numbers, times, etc.).

General method (step-by-step)

• Read the problem carefully and identify the unknown. Choose a variable (say x) to represent it.
• Translate every verbal relation into an algebraic expression in x (pay attention to words like "more", "less", "product", "consecutive", "reciprocal", etc.).
• Form the equation expressing the required condition. Simplify to get a quadratic in standard form: ax^2 + bx + c = 0.
• Solve the quadratic by factoring (if possible), completing the square, or quadratic formula: x = [-b ± sqrt(b^2 − 4ac)]/(2a).
• Check the solutions against the context (e.g. reject negative lengths if the quantity must be positive). State the final answer in words and correct units.

Important modelling tips

• Prefer simple variable choices: for "two consecutive integers" use x and x + 1; for "consecutive even integers" use x and x + 2.
• When equations involve reciprocals, multiply by the common denominator to clear fractions, which usually leads to a quadratic.
• Watch domain restrictions: physical quantities (length, area, time) must be non-negative; sometimes one root is extraneous.

Connection to graphs
A quadratic equation ax^2 + bx + c = 0 corresponds to the x-intercepts of the parabola y = ax^2 + bx + c. Graphical interpretation helps: real distinct roots mean the parabola crosses the x-axis twice; repeated root means tangent; no real roots means no x-intercepts.

What it means: A quadratic equation ax² + bx + c = 0 can be interpreted graphically by considering the quadratic function y = ax² + bx + c. Its graph is a parabola. The roots (solutions) of the quadratic equation are the x-coordinates where the parabola meets the x-axis (y = 0).

Key features to read from the graph:

• Direction: If a > 0 the parabola opens upward (U-shaped); if a < 0 it opens downward (∩-shaped).
• Vertex: The highest or lowest point of the parabola. Coordinates: (h, k) where h = -b/(2a) and k = f(h).
• Axis of symmetry: Vertical line x = -b/(2a) that passes through the vertex.
• x-intercepts (roots): Points where the graph crosses the x-axis. Number of real x-intercepts depends on the discriminant D = b² - 4ac:
• D > 0 → two distinct real roots (two x-intercepts).
• D = 0 → one real repeated root (parabola is tangent to x-axis at the vertex).
• D < 0 → no real roots (parabola does not meet x-axis).
• y-intercept: Point (0, c).

How to sketch from an equation (step-by-step):

1. Write as y = ax² + bx + c.
2. Find y-intercept (0, c).
3. Compute axis of symmetry x = -b/(2a) and vertex (h, k) with k = f(h).
4. Compute discriminant D to know how many x-intercepts to expect; if D ≥ 0 find roots via quadratic formula x = [-b ± √D]/(2a).
5. Plot vertex, intercepts and a few symmetric points about the axis to determine the parabola shape (narrow if |a| large, wide if |a| small).

Interpretation in problem solving: By looking at the graph you can immediately tell how many real solutions a quadratic has, estimate them, and understand maximum/minimum values of y (useful for optimization questions).

Overview: Practice and problem-solving strategies for quadratic equations help you translate word problems into equations, choose efficient solution methods, check answers, and interpret results in context. A systematic approach reduces mistakes and builds confidence.

Step-by-step strategy:

• Read and understand: Identify what is given and what is to be found. Note units and any implicit constraints (e.g., lengths must be positive).
• Introduce a variable: Choose a variable that simplifies relationships (often the unknown quantity itself).
• Form the equation: Translate word relations into algebraic expressions and simplify to standard form ax^2 + bx + c = 0.
• Choose a solving method: Try factorization first (fast if factors are integers); if not obvious, use completing the square or the quadratic formula. Use substitution to reduce complex forms.
• Check discriminant: D = b^2 − 4ac gives the nature of roots. If D < 0, no real solutions (may be invalid for physical problems).
• Solve and interpret: Compute roots, discard extraneous or infeasible roots (negative lengths, etc.), and give answers with correct units/interpretation.
• Verify: Substitute back into the original statement and check reasonableness (signs, approximate magnitudes).

Specific tactics and tips:

• Try to factor by inspection when a, b, c are small integers. Use integer root test (possible rational roots are factors of c over factors of a).
• Use completing the square to find vertex (useful for max/min problems) or to convert to a form that reveals translations: a(x − h)^2 + k.
• Use the quadratic formula when factorization is not easy: x = [−b ± sqrt(b^2 − 4ac)] / (2a).
• Use substitution to convert complicated expressions (e.g., let y = x^2 for biquadratic-like forms) or to reduce word problems.
• For optimisation (max/min) problems, use vertex formula x = −b/(2a) and maximum/minimum value −D/(4a) for y = ax^2 + bx + c.
• Sketch the graph to visualise number of real roots and behaviour (parabola opens up if a > 0, down if a < 0). Graphs help check sign/interval questions.
• Keep units consistent and consider domain restrictions (e.g., time > 0, length > 0).

Common mistakes to avoid:

• Forgetting to simplify to standard form before applying formula.
• Accepting negative or complex roots without checking context.
• Arithmetic errors under the square root; always simplify D before taking sqrt.
• Misplacing the factor a when using vertex formula or completing the square.

Practice advice: Solve a mix of algebraic and word problems: factorable quadratics, those requiring formula, word problems (geometry, motion), and sketching tasks. Time-yourself on some problems to build speed.