Question

What is the discriminant of \[f\left( x \right)=-3{{x}^{2}}-2x-1\]?

Answer 1

Hint: In this problem, we have to find the discriminant for the given equation. We should know that the formula to find the discriminant for the given equation is, \[\Delta ={{b}^{2}}-4ac\], where the equation is of the form \[a{{x}^{2}}+bx+c\]. We can compare the given equation and the general equation to get the value of a, b, c and we then substitute in the determinant formula to get the final answer.

Complete step-by-step answer:
We know that the given equation is,
\[f\left( x \right)=-3{{x}^{2}}-2x-1\]……… (1)
We know that the discriminant formula is,
\[\Delta ={{b}^{2}}-4ac\]
We know that the general formula of the line equation is
\[\Rightarrow a{{x}^{2}}+bx+c\]……. (2)
We can compare the given equation and the general equation to get the value of a, b, c.
By comparing (1) and (2), we get
a = -3, b = -2, c = -1
We can now substitute the values of a, b, c in the formula (2), we get
\[\Rightarrow {{\left( -2 \right)}^{2}}-4\left( -3 \right)\left( -1 \right)=\Delta \]
We can now simplify the above step, we get
\[\Rightarrow \Delta =4-12=-8\]
Therefore, the value of discriminant is \[\Delta =-8\].
Where the discriminant is negative, therefore, we will have complex solutions for this equation.

Note: We should always remember that if the discriminant value is equal to 0 then we have one solution, if the discriminant is positive then we will have two real roots, if the discriminant is negative, then we will have complex numbers as the solution.