Roots of a Quadratic Equation

roots of a quadratic equation

$ax^2+bx+c = 0$ is a quadratic equation in standard form. The degree of this equation is $2$. So, the quadratic equation has two roots.

The two roots are the values of variable of the quadratic equation. Here, $x$ is the variable and the algebraic expression $ax^2+bx+c$ is equal to zero for these two values.

Proof

The two roots of the general form quadratic equation can be derived in mathematics by solving the quadratic equation $ax^2+bx+c = 0$ to extract the variable separately from the equation.

Move the constant term $c$ to right hand side of the equation.

$\implies ax^2 + bx = -c$

Divide both sides by the literal coefficient of the $x^2$.

$\implies \dfrac{ax^2 + bx}{a} = -\dfrac{c}{a}$

$\implies \dfrac{ax^2}{a} + \dfrac{bx}{a} = -\dfrac{c}{a}$

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

$\require{cancel} \implies \Bigg(\dfrac{\cancel{a}}{\cancel{a}}\Bigg)x^2 + \Bigg(\dfrac{b}{a}\Bigg)x = -\dfrac{c}{a}$

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

The second term can be written in the following form.

$\implies x^2 + 1 \times \Bigg(\dfrac{b}{a}\Bigg)x = -\dfrac{c}{a}$

The multiplying factor $1$ can be written as the quotient of $2$ to $2$ because the division of $2$ by $2$ is $1$.

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

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

There is a chance to transform the left hand side expression as a square of an addition based binomial. So, add $\Bigg(\dfrac{b}{2a}\Bigg)^2$ and subtract it by the same term.

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

Move the negative term to right hand side.

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

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

Now, the left hand side expression can be written in square form of a binomial.

$\implies {\Bigg(x + \dfrac{b}{2a}\Bigg)}^2 = \dfrac{b^2}{4a^2} -\dfrac{c}{a}$

$\implies {\Bigg(x + \dfrac{b}{2a}\Bigg)}^2 = \dfrac{b^2 -4ac}{4a^2}$

The square of the left hand side expression becomes square root to right hand side expression.

$\implies x + \dfrac{b}{2a} = \pm \sqrt{\dfrac{b^2 -4ac}{4a^2}}$

$\implies x + \dfrac{b}{2a} = \pm \sqrt{\dfrac{b^2 -4ac}{{(2a)}^2}}$

$\implies x + \dfrac{b}{2a} = \pm \dfrac{\sqrt{b^2 -4ac}}{2a}$

$\implies x = -\dfrac{b}{2a} \pm \dfrac{\sqrt{b^2 -4ac}}{2a}$

$\therefore \,\,\, x = \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}$

Therefore, $x$ is equal to $\dfrac{-b + \sqrt{b^2 -4ac}}{2a}$ and $x$ is also equal to $\dfrac{-b -\sqrt{b^2 -4ac}}{2a}$. For these two values, the quadratic expression is equal to zero.

Representation

If two roots are represented by $\alpha$ and $\beta$, the two roots of the quadratic equation are written as follows.

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

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

But remember, $\alpha$ and $\beta$ are the values of variable $x$ for the quadratic equation $ax^2+bx+c = 0$.


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