Question

What is the discriminant of \[2{x^2} + x - 1 = 0\] and what does that mean?

Answer 1

Hint: In this problem, we have to find the discriminant value of the given quadratic equation and then we have to state the property which is defined by the value of discriminant such as the nature of roots and type of the roots. First of all, we compare the given quadratic equation with the standard form to get the values of \[a,b\] and \[c\] .Then we substitute the values in the formula and find the discriminant. After that based on the nature of the discriminant, we identify the nature and type of the roots.

Formulas used:
The standard form of quadratic equation is:
\[a{x^2} + bx + c = 0\]
Discriminant, \[D = {b^2} - 4ac\]

Complete step by step answer:
Given quadratic equation is:
\[2{x^2} + x - 1 = 0\]
On comparing with the standard quadratic equation i.e., \[a{x^2} + bx + c = 0\]
We get
\[a = 2,{\text{ }}b = 1,{\text{ }}c = - 1\]
Now we know that
\[D = {b^2} - 4ac\]
On substituting the values, we get
\[D = {1^2} - \left( {4 \times 2 \times \left( { - 1} \right)} \right)\]
\[ \Rightarrow D = 1 + 8\]
On adding, we get
\[ \therefore D = 9\]
Thus, the discriminant of the given quadratic equation is \[9\]. Now since our discriminant is positive \[\left( {D > 0} \right)\]. We can say that there are two distinct and real roots. Also, since the discriminant is a perfect square, then the roots will be rational.

Note: Discriminant of a quadratic equation gives us information about the nature of roots of the equation. There are three general cases of the discriminant.
-If the discriminant is positive i.e., \[{b^2} - 4ac > 0\] , then the quadratic equation has real and distinct roots. i.e., there will be two distinct (unequal) roots and they will be real numbers.
-If the discriminant is zero i.e., \[{b^2} - 4ac = 0\] , then the quadratic equation has real and equal roots. i.e., there will be two equal roots and the number will be a real number.
-If the discriminant is negative i.e., \[{b^2} - 4ac < 0\] , then the quadratic equation has no real roots but has complex conjugate roots. i.e., there will be two roots which will be a complex number.
Also notice that we are using the terms “two solutions”. This is because we have given a quadratic equation and the quadratic equation always has two solutions. The degree of the equation indicates the number of solutions.