Question

Number of distinct real roots of the equation \[{{x}^{4}}+4{{x}^{3}}-2{{x}^{2}}-12x+k=0\] is
(A) \[4\text{ if }x\in (-7,9)\]
(B) \[3\text{ if }k=7\]
(C) \[2\text{ if }k<-7\]
(D) \[\text{No root}\text{ if }k>9\]

Answer 1

Hint: The number of roots of the equation depends upon its highest degree. A quadratic equation has the highest degree of two so it will have two roots. The nature of the root depends upon the discriminant of the equation. If we know the concept of the nature of roots then solving this question becomes easier.

Complete step-by-step solution:
A quadratic equation can be solved with the help of three methods and the first method is factorization, the second method is completing the square method and the third one is the discriminant method. The root of the quadratic equation is the value of the unknown variable. If we put the value of the unknown variable in the quadratic equation then the quadratic equation holds for that value.
For the above-given equation in the question, firstly we have to convert it into a quadratic equation through differentiation and then we will find its discriminant. After this, we will find the roots of the quadratic equation.
 In the above question, we have to find the number of distinct real roots of the equation which is shown below.
\[f(x)={{x}^{4}}+4{{x}^{3}}-2{{x}^{2}}-12x+k=0\]
Now we will do the differentiation of the above equation and the following results will be obtained.
\[f'(x)=4{{x}^{3}}+12{{x}^{2}}-4x-12\]
Again we will differentiate the above equation and then the following results will be obtained.
\[f''(x)=12{{x}^{2}}+24x-4\]
Now we will find the discriminant of this equation
\[\begin{align}
  & 12{{x}^{2}}+24x-4=0 \\
 & 4(3{{x}^{2}}+8x-1)=0 \\
 & \Rightarrow 3{{x}^{2}}+8x-1=0 \\
\end{align}\]
The discriminant of the equation will be given by
\[D={{b}^{2}}-4ac\]
Where a, b, c are the coefficients in the equation
\[\begin{align}
  & D={{(8)}^{2}}-4\times 3\times (-1) \\
 & \Rightarrow D=64+12 \\
 & \Rightarrow D=76 \\
\end{align}\]
The roots of this equation can be obtained as shown below
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
The equation is \[3{{x}^{2}}+8x-1\] so its roots are as shown below
\[x=\frac{-8\pm \sqrt{76}}{6}\]
\[\begin{align}
  & \Rightarrow x=\dfrac{-8\pm 2\sqrt{19}}{6} \\
 & \Rightarrow x=\dfrac{-4\pm \sqrt{19}}{3} \\
\end{align}\]
Discriminant of the above equation is positive. so the equation
 \[3{{x}^{2}}+8x-1\] will have two distinct real roots and as this equation is the double differentiation of the equation given in the question and we also know that after differentiation the number of roots of an equation keeps on decreasing.
So the equation \[{{x}^{4}}+4{{x}^{3}}-2{{x}^{2}}-12x+k=0\] will have four distinct real roots.
So the correct option will be an option(A) \[4\text{ }if\text{ }x\in (-7,9)\].

Note: There are various properties of the discriminant. If the discriminant of an equation comes out to be positive then the quadratic equation will have two distinct real roots. If the discriminant equals zero then the equation has a repeated real number solution. If the discriminant comes out to be negative then neither of the solutions obtained will be real numbers.