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.

$ax^2+bx+c = 0$ 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 $x = \{\alpha, \beta\}$.

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

- 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.
- Factorize the expression in the quadratic equation by splitting the middle term.
- Now, equate each factor in $x$ to zero to solve the quadratic equation.

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

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.

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

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

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

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

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

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 $x = \{-3, -5\}$.

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

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