Question

How do you describe the roots of \[4{{x}^{2}}-3x+2=0\]?

Answer 1

Hint: In this problem, we have to find the roots of the given quadratic equation. We can first write the general formula of the quadratic equation and we can compare the general equation to the given quadratic equation to get the value of a, b, c. We can then find the discriminant value to check for the type of roots and we can use the quadratic formula to get the roots value.

Complete step by step solution:
We know that the given quadratic equation is,
\[4{{x}^{2}}-3x+2=0\]……. (1)
We know that the general quadratic equation is,
\[a{{x}^{2}}+bx+c=0\]…….. (2)
We can now compare the equations (1) and (2), we get
a = 4, b = -3, c = 2.
We can now find the discriminant value.
We know that,
\[\Delta ={{b}^{2}}-4ac\]
We can substitute the required values, we get
\[\Rightarrow \Delta ={{\left( -3 \right)}^{2}}-4\left( 4 \right)\left( 2 \right)=-23\]
We can see that, since \[\Delta \] is negative, the quadratic equation has no Real roots. It has a pair of complex roots which are conjugate to one another.
We can now find the roots.
We know that the quadratic formula of the quadratic equation \[a{{x}^{2}}+bx+c=0\] is,
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
We can now substitute the required values in the above formula, we get
\[\begin{align}
  & \Rightarrow x=\dfrac{3\pm \sqrt{\Delta }}{2\left( 4 \right)} \\
 & \Rightarrow x=\dfrac{3\pm \sqrt{-23}}{8} \\
 & \Rightarrow x=\dfrac{3\pm i\sqrt{23}}{8} \\
\end{align}\]
Where \[i\] is the imaginary unit.
Therefore, the complex roots of the given equation \[4{{x}^{2}}-3x+2=0\] is \[x=\dfrac{3}{8}\pm \dfrac{i\sqrt{23}}{8}\] .

Note: Students make mistakes while finding the discriminant value, we should know that the discriminant formula is \[\Delta ={{b}^{2}}-4ac\]. We should also know that negative signs cannot be inside the root, so we assume an imaginary unit \[i\] which gives a complex root value.