Question

How do you solve \[4{{x}^{2}}-2x-1=0\]?

Answer 1

Hint: In order to determine the value of \[x\] in the given equation split the middle terms and then with help of factorising determine the value of \[x\].
Like, for any quadratic equation, \[a{{x}^{2}}+bx+c=0\],
Where, \[a\] and \[b\] are coefficient of \[{{x}^{2}}\] and \[x\] while \[c\] is any constant.
So, the value of roots can be determined by \[\dfrac{-b\pm \sqrt{D}}{2a}\].
Where, \[D\] is nothing but Determinant of the equation whose value is equal to \[{{b}^{2}}-4ac\].
Apply this to determine the value of roots of a given equation.

Complete step by step solution:
As per data given in the question,
As we have to determine the solution of above-mentioned equation,
So,
As per question,
We have,
\[4{{x}^{2}}-2x-1=0\]
As we know that,
General expression of quadratic equation is
\[a{{x}^{2}}+bx+c=0\]
Comparing the given equation with general equation of quadratic equation,
We will get,
\[a=4\], \[b=-2\] and \[c=-1\]
So,
As we know that,
Value of roots will be \[\dfrac{-b\pm \sqrt{D}}{2a}\].
Where, \[D\] is nothing but the determinant of the equation,
Where, \[D={{b}^{2}}-4ac\]
So, determining the value of \[D\].
We will get,
\[D={{\left( -2 \right)}^{2}}-\left[ 4\times \left( 4 \right)\left( -1 \right) \right]\]
\[\Rightarrow D=4+16\]
\[\Rightarrow D=20\]
Now putting the value of \[D\] in above expression,
We will get,
Roots will be,
\[\Rightarrow \]\[\dfrac{-b\pm \sqrt{D}}{2a}\]
So, first root will be \[\dfrac{2+\sqrt{20}}{2\times 4}\]
\[\Rightarrow \]\[\dfrac{2+2\sqrt{5}}{8}\]
While value of other roots will be,
\[\Rightarrow \]\[\dfrac{2-\sqrt{20}}{2\times 4}\]
\[\Rightarrow \]\[\dfrac{2-2\sqrt{5}}{8}\]
Hence, we can easily determine the value of roots of the given equation.

Note: The value of \[D\] or determinant may be positive negative or even zero in some cases,
If the value of \[D\] will be positive then the roots will be real in nature,
If the value of \[D\] is negative then the roots will be real as well as imaginary or simply we can say it will be in the form of a complex number.
While determining the value of \[D\] don’t ignore the negative sign.