Question

For $a{x^2} + bx + c = 0$, if $D > 0$ and $D$ is square of rational numbers and $a, b, c$ and $d$ are rational numbers. Which of the following are true:
(i) Roots are distinct
(ii) Roots are equal
(iii) Roots are rational
(iv) Roots are irrational

A. Only statement (i) is true.
B. Statements (i) and (iii) are true.
C. Only statement (iv) is true.
D. Statements (ii) and (iii) are true.

Answer 1

Hint: In the given question, we are to find the nature of the roots of a quadratic equation $a{x^2} + bx + c = 0$. Quadratic formula can be used to find the roots of a quadratic equation. An equation consisting of the terms with degree two is called a quadratic equation. To find the nature of the roots of the given equation, we have to analyse the discriminant of the equation.

Complete step by step answer:
So, we have, $a{x^2} + bx + c = 0$.
Now, discriminant of the equation $ = D$
Now, we are given $D > 0$.
So, the roots of the quadratic polynomial $a{x^2} + bx + c = 0$ are real and distinct as the discriminant is greater than zero. Also, we are given that D is square of a rational number.We know that the quadratic formula is used to find the values of x for solving the quadratic equation in $x$. So, we have, $a{x^2} + bx + c = 0$

Now, we can find the roots of the equation by plugging in the values of coefficients of the terms.The Quadratic formula is given as:
$x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
where $D$ is the discriminant of the equation.
Now, since the discriminant of an equation is the square of a rational number. So, $\sqrt D $ will have a rational value. Also, we know that addition and division of rational numbers always yield rational numbers. Hence, we get the roots of the quadratic equation as rational.Therefore, statements (i) and (iii) are correct.

Hence, option B is the correct answer.

Note: The discriminant of a quadratic equation tells us about the nature of the roots. The general form of a quadratic equation is given as $a{x^2} + bx + c = 0$. We should know the formula for calculating the discriminant of the equation as $D = {b^2} - 4ac$. The roots of a quadratic equation are rational if the square root of the discriminant of an equation yields a rational value.