Question

1. find the discriminant of $y = 9x^2 - 3x + 2$ and state the type of solution.

discriminant: ____________ types of solution(s): ____
redemption question:
solve for $x$ if $2x + i^2 = 17$. hint: think about what $i^2 =$

Explanation:

Part 1: Find the discriminant and solution type for \( y = 9x^2 - 3x + 2 \)

Step1: Recall discriminant formula

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by \( D = b^2 - 4ac \). Here, \( a = 9 \), \( b = -3 \), \( c = 2 \).

Step2: Calculate discriminant

Substitute \( a = 9 \), \( b = -3 \), \( c = 2 \) into the formula:
\( D = (-3)^2 - 4(9)(2) \)
\( D = 9 - 72 \)
\( D = -63 \)

Step3: Determine solution type

Since \( D < 0 \), the quadratic has two complex (non - real) solutions (conjugate pairs).

Step1: Recall \( i^2 \) value

We know that \( i^2=-1 \) (from the definition of the imaginary unit \( i \), where \( i = \sqrt{-1} \)).

Step2: Substitute \( i^2 \) and solve for \( x \)

Substitute \( i^2=-1 \) into the equation \( 2x + i^2 = 17 \):
\( 2x-1 = 17 \)
Add 1 to both sides: \( 2x=17 + 1=18 \)
Divide both sides by 2: \( x=\frac{18}{2}=9 \)

Answer:

Discriminant: \(-63\)
Types of solution(s): Two complex (non - real) solutions

Part 2: Solve \( 2x + i^2 = 17 \)