4.3. Solution of a Quadratic Equation by Factorisation

Consider the quadratic equation 2 x² – 3x + 1 = 0. If we replace x by 1 on the LHS of this equation, we get (2 × 1²) – (3 × 1) + 1 = 0 = RHS of the equation. We say that 1 is a root of the quadratic equation 2x² – 3x + 1 = 0. This also means that 1 is a zero of the quadratic polynomial 2 x² – 3x + 1.

In this chunk, we see the concept of roots in a quadratic equation. A root of an equation is a value that makes the equation true. Here, substituting x = 1 into the equation validates that the left-hand side equals the right-hand side (zero). Therefore, 1 is acknowledged as a root. We also establish that finding roots is the same as determining the values where the corresponding quadratic polynomial evaluates to zero.

Imagine you are trying to balance a seesaw. The point where it balances is analogous to the root of an equation; just like a seesaw balances at a specific point, an equation holds true at specific x-values.

In general, a real number α is called a root of the quadratic equation ax² + bx + c = 0, a ≠ 0 if a α² + bα + c = 0. We also say that x = α is a solution of the quadratic equation , or that α satisfies the quadratic equation.

This chunk explains that a number α is identified as a root if substituting it back into the equation results in zero. This illustrates not simply a definition but also indicates the relationship between solutions and roots, emphasizing that roots of quadratics also represent the x-values where the polynomial intersects the x-axis.

Think of it as a treasure map: the solutions to the equation are the coordinates where 'X' marks the spot. Just like finding the treasure at those coordinates makes sense, finding roots provides specific values of x that solve the equation.

Note that the zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same. You have observed, in Chapter 2, that a quadratic polynomial can have at most two zeroes. So, any quadratic equation can have at most two roots.

Here, we highlight that both the roots and the zeros refer to the same concept in quadratic equations. Since a quadratic polynomial can have a maximum of two distinct zeroes, any corresponding quadratic equation will also yield up to two roots. This establishes a fundamental property of quadratic equations in mathematics.

Imagine a bus route that has two stops before returning to the station. Those stops represent the two roots or zeroes—just as there can only be at most two designated stops on a given route, a quadratic polynomial is limited to two roots.

You have learnt in Class IX, how to factorise quadratic polynomials by splitting their middle terms. We shall use this knowledge for finding the roots of a quadratic equation.

In this chunk, we link what students learned in previous classes about factorisation to the current topic. The process of factorisation involves rewriting a quadratic equation as a product of its linear factors. By doing so, it becomes easier to identify the roots of the function. This foundational skill from Class IX is crucial as it allows students to solve quadratic equations effectively.

Imagine you are breaking down a complex puzzle into simpler pieces. Each factor represents a piece of the puzzle. When combined correctly, these pieces resolve the main problem of finding the roots of the equation, much like putting together a puzzle reveals the entire image.

Example 3: Find the roots of the equation 2x² – 5x + 3 = 0, by factorisation. Let us first split the middle term – 5x as -2x – 3x....

This example showcases the factorisation technique. In it, we replace -5x with -2x and -3x, making it easier to combine like terms and factor the quadratic polynomial. By setting the factors equal to zero, we can derive the roots of the equation, clearly demonstrating the utility of factorisation in quadratic equations.

Think of this as dissecting a larger task into smaller, manageable segments. Like cooking a complicated dish by breaking it down into basic steps, factorising makes solving the quadratic simpler.

Example 4: Find the roots of the quadratic equation 6x² - x - 2 = 0. Solution: We have... (continued)

This chunk provides further examples where the factorisation method is applied. Students will see how different quadratic equations can be factored similarly to find their roots. By following each step methodically, learners can grasp the importance of practice in mastering factorisation.

Just as a mechanic follows a procedure to fix a car engine, students follow procedures in solving quadratic equations. Each step is crucial to achieving the correct outcome, forging a parallel in understanding.

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