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Question: Find the square roots of \( - 15 - 8i \)...

Find the square roots of −15−8i- 15 - 8i

Explanation

Solution

Hint : Complex number consists of the real part and an imaginary part and is denoted by “Z”. It can be expressed as z=a+ibz = a + ib where “a” is the real part and “b” is the imaginary part. Compare real and imaginary parts and simplify for the resultant values.

Complete step-by-step answer
Given,
z=−15−8iz = - 15 - 8i
Take square-root on both the sides –
z=−15−8iz = \sqrt { - 15 - 8i}
Place, z=a+ibz = a + ib in the above equation –
⇒a+ib=−15−8i\Rightarrow a + ib = \sqrt { - 15 - 8i}
Take square on both the sides of the equations-
⇒(a+ib)2=(−15−8i)2\Rightarrow {\left( {a + ib} \right)^2} = {\left( {\sqrt { - 15 - 8i} } \right)^2}
Expand the brackets by using the sum of two terms whole square identity - (a+b)2=a2+2ab+b2{(a + b)^2} = {a^2} + 2ab + {b^2} . Also, square and square root cancel each other on the right hand side of the equation.
⇒a2+2abi+b2i2=−15−8i\Rightarrow {a^2} + 2abi + {b^2}{i^2} = - 15 - 8i
We know that i2=(−1){i^2} = ( - 1) place in the above equation –
⇒a2+2abi+b2(−1)=−15−8i\Rightarrow {a^2} + 2abi + {b^2}( - 1) = - 15 - 8i
When you multiply one positive term with negative term the resultant value is negative.
⇒a2+2abi−b2=−15−8i\Rightarrow {a^2} + 2abi - {b^2} = - 15 - 8i
The above equation can be re-written as –
⇒a2−b2+2abi=−15−8i\Rightarrow {a^2} - {b^2} + 2abi = - 15 - 8i
Compare the real part a9nd the imaginary part on both sides of the equation.
⇒a2−b2=(−15)\Rightarrow {a^2} - {b^2} = ( - 15) .... (A) and
2ab=(−8)2ab = ( - 8)
Take out common multiple. Common multiple on both the sides of the equation cancels each other.
⇒ab=−4\Rightarrow ab = - 4 .... (B)
We have two equations and two variables-
Take equation (B)
ab=−4 ⇒b=−4a  ab = - 4 \\\ \Rightarrow b = \dfrac{{ - 4}}{a} \\\
Place the above value in equation (A)
⇒a2−(−4a)2=(−15)\Rightarrow {a^2} - {\left( {\dfrac{{ - 4}}{a}} \right)^2} = ( - 15)
Simplify –
⇒a2−16a2=(−15)\Rightarrow {a^2} - \dfrac{{16}}{{{a^2}}} = ( - 15)
Take LCM
⇒a4−16a2=(−15)\Rightarrow \dfrac{{{a^4} - 16}}{{{a^2}}} = ( - 15)
Do-cross multiplication – also place
⇒a4−16=(−15)a2\Rightarrow {a^4} - 16 = ( - 15){a^2}
Take all the terms on one side of the equation-
⇒a4+15a2−16=0\Rightarrow {a^4} + 15{a^2} - 16 = 0
The above equation can be re-written as –
⇒a4+16a2‾−a2−16‾=0\Rightarrow \underline {{a^4} + 16{a^2}} - \underline {{a^2} - 16} = 0
Take the common multiples –

⇒a2(a2+16)−1(a2+16)=0 ⇒(a2+16)(a2−1)=0  \Rightarrow {a^2}({a^2} + 16) - 1({a^2} + 16) = 0 \\\ \Rightarrow ({a^2} + 16)({a^2} - 1) = 0 \\\

We get,
⇒a2=−16 or a2=1\Rightarrow {a^2} = - 16{\text{ or }}{a^2} = 1
Since, a is the real value, we get a=±1a = \pm 1
Place in equation (B)
⇒ab=−4\Rightarrow ab = - 4
When a=1⇒b=−4a = 1 \Rightarrow b = - 4
And When a=−1⇒b=4a = - 1 \Rightarrow b = 4
Hence, the square-roots are –
1+4i and - 1 - 4i1 + 4i{\text{ and - 1 - 4i}}
So, the correct answer is “ 1+4i and - 1 - 4i1 + 4i{\text{ and - 1 - 4i}} ”.

Note : Simplify wisely when finding the roots of the equations. Always remember that the middle term in the perfect square should be equal to the product of the first term and the last term and split the middle term accordingly and keep the signs of the terms double check.