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6
5 (a) Solve the equation
z^(2)-6iz-12=0, giving the answers in the form
x+iy, where
x and
y are real and exact.

[3]

$6$ $\newline$ $5$ (a) Solve the equation $z^{2}-6 \mathrm{i} z-12=0$ , giving the answers in the form $x+\mathrm{i} y$ , where $x$ and $y$ are real and exact. $\newline$ $[3]$

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Full solution

Q.

6

\newline

5

(a) Solve the equation

z^{2}-6 \mathrm{i} z-12=0

, giving the answers in the form

x+\mathrm{i} y

, where

x

and

y

are real and exact.

\newline

[3]

Identify Quadratic Equation: Given the quadratic equation $z^{2} - 6iz - 12 = 0$ , we can solve for $z$ using the quadratic formula $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = -6i$ , and $c = -12$ .
Calculate Discriminant: First, we calculate the discriminant, which is $b^2 - 4ac$ . In this case, it is $(-6i)^2 - 4(1)(-12)$ .
Apply Quadratic Formula: Calculating the discriminant gives us $36i^2 + 48$ . Since $i^2 = -1$ , this simplifies to $-36 + 48$ , which equals $12$ .
Simplify Solutions: Now we can apply the quadratic formula. The solutions for $z$ are $z = \frac{-(-6i) \pm \sqrt{12}}{2 \times 1}$ .
Divide by $2$ : Simplifying the solutions for $z$ gives us $z = \frac{6i \pm 2\sqrt{3}}{2}$ .
Express Solutions in Form: Dividing by $2$ , we get the solutions $z = 3i \pm \sqrt{3}$ .
Express Solutions in Form: Dividing by $2$ , we get the solutions $z = 3i \pm \sqrt{3}$ .We can now express the solutions in the form $x + iy$ . Since there is no real part, $x = 0$ . The solutions are $z = 0 + (3 \pm \sqrt{3})i$ .

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