Solving Quadratic equation by factoring method

A mathematical approach of solving a quadratic equation by transforming the quadratic expression as two factors, is called the factoring method of solving quadratics.

Method

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

$\alpha = \dfrac{-b+\sqrt{b^2 -4ac}}{2a}$ and $\beta = \dfrac{-b-\sqrt{b^2 -4ac}}{2a}$

The sum of the roots of quadratic equation is $\alpha+\beta \,=\, -\dfrac{b}{a}$

The product of the roots of quadratic equation is $\alpha \beta \,=\, \dfrac{c}{a}$

01

The quadratic expression can be adjusted to express the expression in terms of sum of the roots and product of the roots. It can be done by taking literal factor $a$ common from all the terms.

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

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

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

02

Transformation

The literal coefficient of $x$ is $–\dfrac{b}{a}$ and it is the sum of the roots. Similarly, the constant term is $\dfrac{c}{a}$ and it represents the product of the roots of the quadratic equation.

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

03

Factorization

The factorization is only possible if the literal coefficient of x is equal to sum of the roots and the constant term is equal to the product of the roots.

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

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

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

Therefore, the solutions of the quadratic equation are $x = \alpha$ and $x = \beta$.

The quadratic equation $ax^2+bx+c = 0$ is transformed in factored form $a(x-\alpha)(x-\beta) = 0$. Now, it is used to find the roots of quadratic equation. Hence, the mathematical approach is called the factoring method of solving quadratic equation.

Example

$x^2+8x+15 = 0$ is an example quadratic equation.

Firstly consider the constant term and it is $15$. Try to express it as factors of two numbers. Possibly, $15 \times 1 = 15$ and $5 \times 3 = 15$.

Add the two numbers, $15+1 = 16$ and $5+3 = 8$. The sum of the numbers $15$ and $1$ is not equal to the literal coefficient of $x$ but the sum of the numbers $5$ and $3$ is the literal coefficient of the $x$.

Therefore, write the literal coefficient of $x$ as the sum of the numbers $5$ and $3$ and also write the constant term as the product of the numbers $5$ and $3$.

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

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

Express the quadratic equation in factoring form.

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

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

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

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