Math Doubts

Solving the Quadratic equations by factoring

A mathematical approach of factoring the expression of a quadratic equation for solving it is called the method of solving the quadratic equation by factorization (or) factorisation.

Introduction

$ax^2+bx+c = 0$ is a quadratic equation in general form. If $\alpha$ and $\beta$ represent the solutions of this equation, then

$(1) \,\,\,$ The sum of the roots is $\alpha+\beta \,=\, -\dfrac{b}{a}$

$(2) \,\,\,$ The product of the roots is $\alpha \beta \,=\, \dfrac{c}{a}$

Now, let us try to express the quadratic equation in terms of the sum and product of the roots.

$\implies$ $a\Bigg[x^2+\dfrac{bx}{a}+\dfrac{c}{a}\Bigg] = 0$

$\implies$ $a\Bigg[x^2+\Bigg(\dfrac{b}{a}\Bigg)x+\dfrac{c}{a}\Bigg] = 0$

$\implies$ $a\Bigg[x^2-\Bigg(\dfrac{-b}{a}\Bigg)x+\dfrac{c}{a}\Bigg] = 0$

The literal coefficient of $-x$ is $–\dfrac{b}{a}$, which is sum of the roots of the quadratic equation. The constant term is $\dfrac{c}{a}$, which is the product of the roots of the quadratic equation. Now, substitute them in the simplified equation.

$\implies$ $a [x^2-(\alpha+\beta)x+\alpha \beta ] = 0$

The expression $a [x^2-(\alpha+\beta)x+\alpha \beta ]$ in the simplified quadratic equation can be factored by splitting the middle term.

$\implies a[x^2 – \alpha x -\beta x + \alpha \beta] = 0$

$\implies a[x(x-\alpha) -\beta(x-\alpha)] = 0$

$\,\,\, \therefore \,\,\,\,\,\, a(x-\alpha)(x-\beta) = 0$

In this case, $a \ne 0$. So, $x-\alpha = 0$ and $x-\beta = 0$. Therefore, $x = \alpha$ and $x = \beta$ are the solutions of the quadratic equation. The solution of the quadratic equation is $x = \{\alpha, \beta\}$.

Method

The quadratics can be solved by factoring in three simple steps.

  1. Simplify the quadratic equation in such a way that one side of the equation is in mathematical form and the other side of the equation is equal to zero.
  2. Factorize the expression in the quadratic equation by splitting the middle term.
  3. Now, equate each factor in $x$ to zero to solve the quadratic equation.

Example

Solve $x^2 = -(8x+15)$

Step – 1

Let’s set the equation in such a way that the left side of the equation is an expression in mathematical form and the right side of the equation is equal to zero.

$\implies$ $x^2+8x+15 = 0$

Step – 2

The polynomial in the left hand side of the equation can be factored using the method of factorization by splitting the middle term.

$\implies$ $x^2+3x+5x+3 \times 5 = 0$

$\implies$ $x(x+3)+5(x+3) = 0$

$\implies$ $(x+3)(x+5) = 0$

Step – 3

Now, solve the quadratic equation by equating each factor that contains a variable to zero.

$x+3 = 0$ and $x+5 = 0$. Therefore, $x = -3$ and $x = -5$ are solutions of this quadratic equation by factoring method and the roots are $x = \{-3, -5\}$.

Worksheet

List of problems on quadratic equations with solutions to learn and practice solving the quadratic equations by factoring.

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