Solving Quadratic Equations by Factorisation

Factorisation of a quadratic equation means expressing the quadratic polynomial ax² + bx + c as a product of two linear factors. If we can write:

ax² + bx + c = (px + q)(rx + s)

then the equation ax² + bx + c = 0 becomes (px + q)(rx + s) = 0. By the zero product property (if the product of two factors is zero, at least one factor must be zero), we get:

px + q = 0 → x = -q/p
rx + s = 0 → x = -s/r

These are the two roots of the quadratic equation.

The zero product property states: if A × B = 0, then A = 0 or B = 0 (or both). This property holds for real numbers and is the cornerstone of the factorisation method. It is crucial to understand that this property works ONLY when the product equals zero — if A × B = 6, we cannot conclude that A = 6 or B = 6.

Factorisation is most efficient when the roots are rational numbers, which happens when the discriminant D = b² - 4ac is a perfect square. If D is not a perfect square, the roots are irrational and factorisation into linear factors with rational coefficients is not possible — in such cases, the quadratic formula or completing the square should be used.

Detailed Step-by-Step Process:

Step 1 — Standard Form: Ensure the equation is in the form ax² + bx + c = 0. If not, rearrange.

Step 2 — Compute ac: Multiply the coefficient of x² by the constant term. Note the sign.

Step 3 — Find the Split: Look for two numbers whose product is ac and whose sum is b. This is the most important step. Consider all factor pairs of ac (including negative ones) and check which pair sums to b.

Step 4 — Rewrite: Replace bx with the sum of the two terms found in Step 3.

Step 5 — Group and Factor: Group the four terms into two pairs. Take out the common factor from each pair. You should get two brackets that are identical.

Step 6 — Write as Product: Express as a product of two linear factors.

Step 7 — Solve: Set each factor equal to zero and solve for x.

How to Find the Right Pair in Step 3:

• List all factor pairs of |ac|.
• If ac is positive, both numbers have the same sign (both positive if b > 0, both negative if b < 0).
• If ac is negative, the numbers have opposite signs. The larger magnitude number takes the sign of b.

Special Cases:

• When c = 0: ax² + bx = 0 → x(ax + b) = 0. One root is always x = 0.
• When b = 0: ax² + c = 0. This is a difference of squares if c < 0: ax² = |c| → x² = |c|/a → x = ± √(|c|/a).
• Perfect square trinomial: ax² + bx + c = (√a · x + √c)² when b = 2√(ac).

The factorisation method for solving quadratic equations has widespread applications:

Number Theory: Finding numbers that satisfy given conditions about products, sums, and differences often leads to quadratic equations that are best solved by factorisation. For example, finding two consecutive integers whose product is 132 leads to x(x+1) = 132, or x² + x - 132 = 0, which factors as (x - 11)(x + 12) = 0.

Geometry: Problems involving areas and dimensions of rectangles, triangles, and other shapes frequently produce quadratic equations with integer or rational roots that factorise neatly. For instance, finding the dimensions of a rectangle with area 60 cm² and perimeter 34 cm leads to a factorisable quadratic.

Physics: Kinematic equations (involving distance, velocity, and acceleration) sometimes yield quadratic equations. When a ball is thrown upward with initial velocity 20 m/s, the equation h = 20t - 5t² = 0 factors as 5t(4 - t) = 0, giving t = 0 or t = 4 seconds. The factorisation reveals both the starting and landing times.

Finance: Compound interest and profit/loss problems occasionally lead to quadratic equations. Factorisation gives exact answers without rounding errors. For example, if a sum doubles in n years at a certain rate, the resulting quadratic often has rational roots.

Computer Science: Algorithm analysis sometimes involves solving recurrence relations that reduce to quadratic equations. Factorisation provides clean closed-form solutions. The Fibonacci recurrence, when solved via its characteristic equation x² - x - 1 = 0, illustrates this connection.

Daily Life: Planning dimensions (garden beds, tiles, rooms) subject to area constraints leads to quadratic equations. The factorisation method gives practical, exact dimensions. For example, tiling a floor where the number of tiles along one side is 3 more than along the other, with a total of 108 tiles, gives a quadratic equation that factors neatly.

Age Problems: Problems where the product of ages or the relationship between current and future ages leads to quadratic equations are commonly solved by factorisation in school-level mathematics.