Question

How do you determine if a solution to a quadratic equation is rational or irrational by using the discriminant?

Answer 1

Hint: Discriminant of the quadratic equation is generally denoted by “D” or $\Delta $ which is used to solve the second degree equations. Here we will determine using the standard quadratic equation. The discriminant of the polynomial equation is the value computed from the coefficients that helps us determine the type of roots it has, specifically whether which are real or non-real and distinct or repeated.

Complete step-by-step solution:
Consider the general form of the quadratic equation.
$a{x^2} + bx + c = 0$
The solution for the above quadratic equation can be given by:
$x = {{ - b \pm \sqrt \Delta }}{{2a}}$ … (A)
Where, Discriminant $\Delta = {b^2} - 4ac$
So, here the discriminant can be used to characterize the solutions of the equation as:
i) $\Delta > 0$
For that we get solutions two separate real solutions
$x = {{ - b + \sqrt \Delta }}{{2a}}$ and $x = {{ - b - \sqrt \Delta }}{{2a}}$
ii) $\Delta = 0$
Place in the equation (A)
$ \Rightarrow x = {{ - b}}{{2a}}$
Which gives two coincident real solutions or one repeated root.
iii) $\Delta < 0$
Discriminant being imaginary, we get two imaginary solutions.
$x = {{ - b + i\sqrt \Delta }}{{2a}}$ and $x = {{ - b - i\sqrt \Delta }}{{2a}}$
Which gives no real solutions. It has the complex conjugate pair of non-real roots.

Additional information: A quadratic equation is an equation of second degree, it means at least one of the terms is squared. Standard equation is $a{x^2} + bx + c = 0$ where a,b,and c are constant and “a '' can never be zero and “x” is unknown. Constants are the terms with fixed value such as the numbers it can be positive or negative whereas the variables are terms which are denoted by small alphabets such as x, y, z, a, b, etc. Be careful while moving any term from one side to another.

Note: The discriminant is also known as the delta. The delta is a square number then the quadratic will factorize since the square root in the quadratic formula would be the rational.