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

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 $x$ .