Chapter 9.5, Problem 37E

a.

To determine

To solve:The equation

Answer to Problem 37E

  $\pm 3-4i$ is the solution.

Explanation of Solution

Given:

  $x^2+8ix-25=0$

Concept Used:

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation.

Calculation:

For quadratic equation

  $a{x}^2+bx+c=0$

Solution, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here $a=1,b=8i,c=-25$

  $\begin{array}{l}x=\frac{-2i\pm \sqrt{{(8i)}^2-4(1)(-25)}}{2(1)}\\ \Rightarrow \frac{-8i\pm \sqrt{64i^2+100}}{2}\\ \Rightarrow \frac{-8i\pm \sqrt{-64+100}}{2}\\ \Rightarrow \frac{-8i\pm 6}{2}=\pm 3-4i\end{array}$

Hence,

  $\pm 3-4i$ is the solution.

b.

To determine

To find: The solution of equationare whether complex conjugates.

Answer to Problem 37E

The solution of equationare not complex conjugates.

Explanation of Solution

Given:

  $x^2+8ix-25=0$

Concept Used:

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude, but opposite in sign.

Calculation:

For quadratic equation

  $a{x}^2+bx+c=0$

Solution, $x=\pm 3-4i$

This solution are two complex numbers with opposite real components, while the imaginary components are equal.

(For conjugate numbers the form is $a\pm bi$ where real components equal & imaginary components opposite signs)

Hence,

The solution of equationare not complex conjugates.

c.

To determine

To find:Complex solution of the equation.

Answer to Problem 37E

The solutions need not be complex conjugates (as we see the case here).

Explanation of Solution

Given:

  $x^2+8ix-25=0$

Concept Used:

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation.

Calculation:

For quadratic equation

  $a{x}^2+bx+c=0$

Given equation $x^2+8ix-25=0$

Solution, $x=\pm 3-4i$

What we know about a quadratic equation with real coefficient is that it will have either

1. Two real solutions, or
2. Same real solution, or
3. Complex (conjugate) solutions.

Here equation does not have all real coefficient.

Hence,

The solutions need not be complex conjugates (as we see the case here).

d.

To determine

To check: The solution

Answer to Problem 37E

The solution is correct.

Explanation of Solution

Given:

  $x^2+8ix-25=0$

Concept Used:

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation.

Calculation:

For quadratic equation

Given equation $x^2+8ix-25=0$

Solution, $x=\pm 3-4i$

For $x=3-4i$

  $\begin{array}{l}{(}^3+8i(3-4i)-25\\ \Rightarrow (9-24i+16i^2)+24i-32i^2-25\\ \Rightarrow (9-16+32-25)\\ \Rightarrow 0\end{array}$

Same for $x=-3-4i$

Hence,

The solution is correct.