Question

If \[\sqrt { - 7 + 24i} \] is \[x + iy\] then \[x\] is equal to
A. \[ \pm 1\]
B. \[ \pm 2\]
C. \[ \pm 3\]
D. \[ \pm 4\]

Answer 1

Hint:A complex number is a number that can be expressed in the form \[x + iy\] where \[x\] and \[y\] are real numbers and \[i\] s a symbol called the imaginary unit , and satisfying the equation \[{i^2} = - 1\] . Because no "real" number satisfies this equation \[i\] was called an imaginary number . for a complex number \[x + iy\] , \[x\] is called the real part and \[y\] is called the imaginary part.

Complete step by step answer:
We are given that \[\sqrt { - 7 + 24i} = x + iy\]
Now on squaring both the sides we get
\[ - 7 + 24i = {\left( {x + iy} \right)^2}\]
Now solving the right hand side of the equation we get
\[ - 7 + 24i = {x^2} + {i^2}{y^2} + 2xyi\]
Now since \[{i^2} = - 1\]
Therefore the above equation becomes
\[ - 7 + 24i = {x^2} - {y^2} + 2xyi\]
On comparing the real and imaginary parts we get
\[ - 7 = {x^2} - {y^2}\] and \[24 = 2xy\]
Therefore we get
 \[ - 7 = {x^2} - {y^2}\] and \[12 = xy\]
Taking the value of \[y\] in terms of \[x\] from the equation \[12 = xy\] we get
\[y = \dfrac{{12}}{x}\]
Now putting this value of \[y\] in the equation \[ - 7 = {x^2} - {y^2}\] we get
\[ - 7 = {x^2} - {\left( {\dfrac{{12}}{x}} \right)^2}\]
On simplification we get
\[ - 7 = {x^2} - \dfrac{{144}}{{{x^2}}}\]
On taking the LCM on the right hand side of the equation we get
\[ - 7 = \dfrac{{{x^4} - 144}}{{{x^2}}}\]
On cross multiplication we get
\[ - 7{x^2} = {x^4} - 144\]
Taking all the terms on one side we get
\[{x^4} + 7{x^2} - 144 = 0\]
This is a quadratic equation in variable \[{x^2}\]
We know that general form of a quadratic equation in the variable \[x\] is of the form \[a{x^2} + bx + c = 0\]
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Hence by using the quadratic formula to find the roots of a quadratic equation we have
\[{x^2} = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
In particular
\[{x^2} = \dfrac{{ - 7 \pm \sqrt {{{(7)}^2} - 4(1)( - 144)} }}{{2(1)}}\]
On simplification we get
\[{x^2} = \dfrac{{ - 7 \pm \sqrt {49 + 576} }}{2}\]
Which further simplifies to
\[{x^2} = \dfrac{{ - 7 \pm \sqrt {625} }}{2}\]
Which further simplifies to
\[{x^2} = \dfrac{{ - 7 \pm 25}}{2}\]
Therefore we have
\[{x^2} = \dfrac{{ - 7 + 25}}{2}\] or \[{x^2} = \dfrac{{ - 7 - 25}}{2}\]
Which gives us
\[{x^2} = 9\] or \[{x^2} = - 16\]
Now we know that \[{x^2} = - 16\] is not possible because the square of any number cannot be equal to a negative number.
Therefore we get \[{x^2} = 9\]
This gives us \[x = \pm 3\]

Therefore option C is the correct answer.

Note: A complex number is a number that can be expressed in the form \[x + iy\] where \[x\] and \[y\] are real numbers and \[i\] s a symbol called the imaginary unit.Remember the quadratic formula. Do not miss any possible value of \[x\] .