Answer:
[tex]\bold{x=\dfrac{-3\pm i\sqrt{7}}{2}}[/tex]
Step-by-step explanation:
Given quadratic equation is:
[tex]-x^2 - 3x = 4[/tex]
Rewriting the given equation:
[tex]-x^2 - 3x - 4 = 0[/tex]
OR
[tex]x^2 + 3x + 4=0[/tex]
Solution of a quadratic equation represented as [tex]ax^2+bx+c=0[/tex] is given as:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Comparing the given equation with standard equation:
a = 1
b = 3
c = 4
So, the roots are:
[tex]x=\dfrac{-3\pm\sqrt{3^2-4\times 1 \times 4}}{2\times 1}\\\Rightarrow x=\dfrac{-3\pm\sqrt{9-16}}{2}\\\Rightarrow x=\dfrac{-3\pm\sqrt{-7}}{2}[/tex]
[tex]\sqrt{-7 }[/tex] can be written as [tex]7\sqrt{-1}[/tex]
and [tex]\sqrt{-1} = i[/tex]
So, [tex]\sqrt{-7} = i\sqrt7[/tex]
The numbers containing [tex]i[/tex] in them, are called as complex numbers.
Therefore, the roots of the equation can be written as:
[tex]\Rightarrow \bold{x=\dfrac{-3\pm i\sqrt{7}}{2}}[/tex]
