Question

Find the square root of the following complex number $7 + 24i.$

Answer 1

Hint: We can assume the square root of our complex number is equal to some other complex number. Then by comparing the real part in L.H.S with a real part in R.H.S and the imaginary part in L.H.S with an imaginary part in R.H.S, we can calculate the square root. There may be multiple values.

Complete step-by-step answer:
We know that the square root of $7 + 24i.$ is $\sqrt {7 + 24i} $.
Now, let’s assume
$\sqrt {7 + 24i} = x + iy$
Squaring both sides,
$7 + 24i = {\left( {x + iy} \right)^2}$
Using formula, ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
$7 + 24i = {x^2} + {\left( {iy} \right)^2} + 2xyi$
We know that value of ${i^2}$ is $ - 1\,.$
On putting the value, we get
$7 + 24i = {x^2} - {y^2} + 2xyi$
Now, comparing the real part in L.H.S with a real part in R.H.S and imaginary part in L.H.S with an imaginary part in R.H.S.
On comparing imaginary part, we get
$24 = 2xy$
On dividing, we get
$12 = xy$
$y = \dfrac{{12}}{x}$
On comparing real part, we get
${x^2} - {y^2} = 7$
Putting the above value of y
${x^2} - {\left( {\dfrac{{12}}{x}} \right)^2} = 7$
On squaring, we get
${x^2} - \dfrac{{144}}{{{x^2}}} = 7$
Taking L.C.M
$\dfrac{{{x^4} - 144}}{{{x^2}}} = 7$
On cross-multiplication, we get
${x^4} - 144 = 7{x^2}$
Transposing $7{x^2}$ to L.H.S
${x^4} - 7{x^2} - 144 = 0$
We can also write $7{x^2} = 16{x^2} - 9{x^2}$
${x^4} - \left( {16{x^2} - 9{x^2}} \right) - 144 = 0$
${x^4} - 16{x^2} + 9{x^2} - 144 = 0$
Taking common ${x^2}\,\,and\,\,9$
${x^2}\left( {{x^2} - 16} \right) + 9\left( {{x^2} - 16} \right) = 0$
Now, taking ${x^2} - 16$ common
$\left( {{x^2} - 16} \right)\left( {{x^2} + 9} \right) = 0$
${x^2} + 9$ can never be zero because ${x^2}$ is a positive number and by adding a positive number with another positive number we get another positive number. Therefore, only one case exists here.
${x^2} - 16 = 0$
${x^2} = 16$
$x = \pm 4$
Also, we know that
$y = \dfrac{{12}}{x}$
Now, putting the value of x
$y = \pm \dfrac{{12}}{4}$
On division, we get
$y = \pm 3$
Therefore, $\sqrt {7 + 24i} = \pm 4 \pm 3i$
Hence, the required square root of our complex number is $ \pm 4 \pm 3i$.

Note: The method given in the question is a standard method for finding the square root of a complex number. If the given complex number is $a + ib$ and the required complex number is $x + iy$ then we can conclude that $a = {x^2} - {y^2}$ and $b = 2xy$.