Question

lesson 5.1 exit ticket

1. what is ( i ) equal to?

1. what is ( i^2 ) equal to?

1. solve ( 3x^2 + 75 = 0 )

1. what is the product of ( (3 - 2i)(-7 - i) )? write your answer in simplest ( a + bi ) form.

Explanation:

Question 1

Step1: Recall imaginary unit definition

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \).

Step1: Square the imaginary unit

Using the definition of \( i = \sqrt{-1} \), we square both sides: \( i^2 = (\sqrt{-1})^2 \). By the property of square roots, \( (\sqrt{a})^2 = a \) for \( a\geq0 \), here \( a = - 1 \), so \( i^2=-1 \).

Step1: Isolate \( x^2 \) term

Start with the equation \( 3x^2 + 75 = 0 \). Subtract 75 from both sides: \( 3x^2= - 75 \). Then divide both sides by 3: \( x^2=\frac{-75}{3}=-25 \).

Step2: Solve for \( x \)

Take the square root of both sides. Since \( x^2=-25 \), we use the imaginary unit. \( x = \pm\sqrt{-25}=\pm\sqrt{25}\times\sqrt{-1}=\pm5i \).

Answer:

\( i = \sqrt{-1} \)

Question 2