Nature of Roots of a Quadratic Equation for Square discriminant
Fact-checked:
The roots of a quadratic equation are real and rational if the discriminant of a quadratic equation is positive and a perfect square.
Introduction
Algebraically, a quadratic equation is written as $ax^2+bx+c \,=\, 0$ in mathematical form. Now, the discriminant ($\Delta$ or $D$) of this quadratic equation is written as $b^2-4ac$.
The zeros or roots of the quadratic equation in terms of discriminant are written in the following two forms.
$(1).\,\,\,$ $\dfrac{-b+\sqrt{\Delta}}{2a}$
$(2).\,\,\,$ $\dfrac{-b-\sqrt{\Delta}}{2a}$
When the discriminant of the quadratic equation is positive, the square root of the discriminant will be positive too.
For example $\sqrt{\Delta} \,=\, d$
The zeros or roots of the quadratic equation can be expressed in the following two mathematical forms.
$(1).\,\,\,$ $\dfrac{-b+d}{2a}$
$(2).\,\,\,$ $\dfrac{-b-d}{2a}$
The two roots clearly express that the roots of the quadratic equation are real and distinct.
Example
The following quadratic equation is an example for a quadratic equation which contains rational numbers as coefficients and the discriminant is positive number, can expressed as a perfect square number.
$3x^2-10x+3 = 0$
$3$, $-10$ and $3$ are rational numbers and they are coefficients of this quadratic equation. Now, find the discriminant of the quadratic equation.
$\Delta = {(-10)}^2-4 \times 3 \times 3$
$\implies \Delta = 100-36$
$\implies \Delta = 64$
$\implies \Delta = 8^2$
The discriminant is expressed as a square of a number. Now, evaluate the roots of this quadratic equation.
$x \,=\, \dfrac{-(-10) \pm \sqrt{{(-10)}^2-4 \times 3 \times 3}}{2 \times 3}$
$\implies$ $x \,=\, \dfrac{10 \pm \sqrt{64}}{6}$
$\implies$ $x \,=\, \dfrac{10 \pm \sqrt{8^2}}{6}$
$\implies$ $x \,=\, \dfrac{10 \pm 8}{6}$
$\implies$ $x \,=\, \dfrac{10+8}{6}$ and $\dfrac{10-8}{6}$
$\implies$ $x \,=\, \dfrac{18}{6}$ and $\dfrac{2}{6}$
$\implies$ $\require{cancel} x \,=\, \dfrac{\cancel{18}}{\cancel{6}}$ and $\dfrac{\cancel{2}}{\cancel{6}}$
$\,\,\, \therefore \,\,\,\,\,\,$ $x \,=\, 3$ and $x \,=\, \dfrac{1}{3}$
The roots of the quadratic equation $3x^2-10x+3 = 0$ are $3$ and $\dfrac{1}{3}$, they are real numbers but rational because the discriminant is a perfect square of a number.
Therefore, it is proved that the roots of a quadratic equation are distinct rational numbers when the discriminant is a positive number, which can be expressed as a square of another number.
