Question

What is the discriminant of a quadratic function?

Answer 1

Hint: The discriminant is the part of the quadratic formula under the square root. Quadratic formula is used to find the roots of the quadratic equation. Quadratic equations are the polynomial equations of degree 2 in one variable. There will be 2 roots for the quadratic equation. The discriminant can be positive, zero, or negative, and this determines the number of solutions of a given quadratic equation.

Complete step by step solution:
To understand deeply on discriminants, let us first see what a quadratic equation is. Quadratic equations are the polynomial equations of degree 2 in one variable. We can write the standard form of quadratic equation as $f\left( x \right)=a{{x}^{2}}+bx+c$ , where a, b and c are real numbers and $a\ne 0$ . We can see that this quadratic equation will have two roots since the degree is 2. To find the roots, we will use the quadratic formula. We can write the quadratic formula as
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
The discriminant is the part of the quadratic formula under the square root. Hence, we can write the discriminant as
$D=\sqrt{{{b}^{2}}-4ac}$
The discriminant can be positive, zero, or negative, and this determines the number of solutions of a given quadratic equation.
Let us consider a positive discriminant, that is, $D>0$ . This indicates that the quadratic equation has two distinct real number solutions.
Now, let us consider a zero discriminant, that is, $D=0$ . This indicates that the quadratic equation has a repeated real number solution.
Now, if the discriminant is negative, that is, $D<0$ , then we can say that the quadratic equation will have neither of their solutions as real numbers, that is, the solutions will be imaginary.

Note: Students must know the standard form of quadratic equations to find the discriminants and the roots. The $\pm $ sign will be split to + and – signs to find the two roots. Students must know the properties of discriminant to get a clear view of the type of roots.