Math Doubts

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

Basic adjustment

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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