Question
Find the square root of 3 - 4i
✓
Answer
Let $x + yi =\sqrt{3-4 i}$
Squaring both sides, we get
$x^2-y^2+2 x y i=3-4 i$
Equating the real and imaginary parts
$x^2-y^2=3 \ldots . \text { (i) }$
and $2 x y=-4 \Rightarrow x y=-2$
Now from the identity, we know
$\begin{array}{l}\left(x^2+y^2\right)=\left(x^2-y^2\right)^2+4 x^2 y^2 \\
\left(x^2+y^2\right)^2=(3)^2+4(-2)^2 \\
=9+16=25\end{array}$
$\therefore x^2+y^2=5 \ldots$ (ii) [Neglecting (-) sign as $\left.x^2+y^2>0\right]$
Solving (i) and (ii) we get
$\begin{array}{l} x ^2=4 \text { and } y ^2=1 \\
x= \pm 2 \text { and } y= \pm 1\end{array}$
Since the sign of xy is negative
$\begin{array}{l}\therefore \text { if, } x =2, y =-1 \\
\text { and if } x =-2, y =1 \\
\therefore \sqrt{-5+12 i}= \pm(2-i)\end{array}$
Squaring both sides, we get
$x^2-y^2+2 x y i=3-4 i$
Equating the real and imaginary parts
$x^2-y^2=3 \ldots . \text { (i) }$
and $2 x y=-4 \Rightarrow x y=-2$
Now from the identity, we know
$\begin{array}{l}\left(x^2+y^2\right)=\left(x^2-y^2\right)^2+4 x^2 y^2 \\
\left(x^2+y^2\right)^2=(3)^2+4(-2)^2 \\
=9+16=25\end{array}$
$\therefore x^2+y^2=5 \ldots$ (ii) [Neglecting (-) sign as $\left.x^2+y^2>0\right]$
Solving (i) and (ii) we get
$\begin{array}{l} x ^2=4 \text { and } y ^2=1 \\
x= \pm 2 \text { and } y= \pm 1\end{array}$
Since the sign of xy is negative
$\begin{array}{l}\therefore \text { if, } x =2, y =-1 \\
\text { and if } x =-2, y =1 \\
\therefore \sqrt{-5+12 i}= \pm(2-i)\end{array}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Show the following quadratic equation by factorization method:
$\text{x}^2 +2\text{ x} +\frac{3}{2}=0$
→$\text{x}^2 +2\text{ x} +\frac{3}{2}=0$
Differentiate the following functions.
$\frac{\text{x}^{5}-\cos\text{x}}{\sin\text{x}}$
→$\frac{\text{x}^{5}-\cos\text{x}}{\sin\text{x}}$
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, Find the first term.
A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?
Find the linear inequalities for which the shaded region in the given figure is the solution set.
In drilling world’s deepest hole it was found that the temperature T in degree celcius, x km below the earth’s surface was given by $\text{T}=30^\circ+25^\circ(\text{x}-3), 3\leq\text{x}\leq15$At what depth will the temperature be between 155°C and 205°C?
→Find the sum of the following arithmetic progression:
$\frac{\text{x}-\text{y}}{\text{x}+\text{y}},\ \frac{3\text{x}-2\text{y}}{\text{x}+\text{y}},\ \frac{5\text{x}-3\text{y}}{\text{x}+\text{y}},\ ...$ to n terms.
→$\frac{\text{x}-\text{y}}{\text{x}+\text{y}},\ \frac{3\text{x}-2\text{y}}{\text{x}+\text{y}},\ \frac{5\text{x}-3\text{y}}{\text{x}+\text{y}},\ ...$ to n terms.
The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1 . Find the common ratio and the terms.
→Find the length of the line-segment joining the vertex of the parabola y2 = 4ax and a point on the parabola where the line-segment makes an angle $\theta$ to the x-axis.
→Find four numbers forming a geometric progression in which the third term is greater than the first term by 9 and the second term is greater than by 4th by 18.