Educational worksheet on finding the discriminant of quadratic equations, promoting diversity in STEM.

The image shows a worksheet titled "Finding Discriminant", which is a mathematical task involving quadratic equations. The goal is to calculate the discriminant for each given quadratic equation. Below, I will explain how to solve this problem step by step.

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What is the Discriminant?

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[
\Delta = b^2 - 4ac
\]
The discriminant helps determine the nature of the roots of the quadratic equation:
1. If \( \Delta > 0 \), the equation has two distinct real roots.
2. If \( \Delta = 0 \), the equation has one repeated real root (a double root).
3. If \( \Delta < 0 \), the equation has two complex conjugate roots.

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Steps to Solve Each Problem

For each quadratic equation provided in the worksheet, follow these steps:
1. Identify the coefficients \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \).
2. Substitute these values into the discriminant formula: \( \Delta = b^2 - 4ac \).
3. Simplify the expression to find the value of the discriminant.

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Example Problems from the Image

Let's solve a few examples from the worksheet:

#### 1. \( 2x^2 + 5x + 6 = 0 \)
- Coefficients: \( a = 2 \), \( b = 5 \), \( c = 6 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 5^2 - 4(2)(6) = 25 - 48 = -23
\]
- Result: \( \Delta = -23 \)

#### 2. \( x^2 - 4x + 4 = 0 \)
- Coefficients: \( a = 1 \), \( b = -4 \), \( c = 4 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0
\]
- Result: \( \Delta = 0 \)

#### 3. \( 7x^2 + 6 = 0 \)
- Rewrite the equation as \( 7x^2 + 0x + 6 = 0 \)
- Coefficients: \( a = 7 \), \( b = 0 \), \( c = 6 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 0^2 - 4(7)(6) = 0 - 168 = -168
\]
- Result: \( \Delta = -168 \)

#### 4. \( x^2 - 3x + 4 = 0 \)
- Coefficients: \( a = 1 \), \( b = -3 \), \( c = 4 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-3)^2 - 4(1)(4) = 9 - 16 = -7
\]
- Result: \( \Delta = -7 \)

#### 5. \( 3x^2 + 6x + 6 = 0 \)
- Coefficients: \( a = 3 \), \( b = 6 \), \( c = 6 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 6^2 - 4(3)(6) = 36 - 72 = -36
\]
- Result: \( \Delta = -36 \)

#### 6. \( 9y^2 - 8y - 1 = 0 \)
- Coefficients: \( a = 9 \), \( b = -8 \), \( c = -1 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = (-8)^2 - 4(9)(-1) = 64 + 36 = 100
\]
- Result: \( \Delta = 100 \)

#### 7. \( 5x^2 = 0 \)
- Rewrite the equation as \( 5x^2 + 0x + 0 = 0 \)
- Coefficients: \( a = 5 \), \( b = 0 \), \( c = 0 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 0^2 - 4(5)(0) = 0 - 0 = 0
\]
- Result: \( \Delta = 0 \)

#### 8. \( 12x^2 + 2x = 0 \)
- Rewrite the equation as \( 12x^2 + 2x + 0 = 0 \)
- Coefficients: \( a = 12 \), \( b = 2 \), \( c = 0 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 2^2 - 4(12)(0) = 4 - 0 = 4
\]
- Result: \( \Delta = 4 \)

#### 9. \( 6d^2 - 5 = 0 \)
- Rewrite the equation as \( 6d^2 + 0d - 5 = 0 \)
- Coefficients: \( a = 6 \), \( b = 0 \), \( c = -5 \)
- Discriminant:
\[
\Delta = b^2 - 4ac = 0^2 - 4(6)(-5) = 0 + 120 = 120
\]
- Result: \( \Delta = 120 \)

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Final Answers

Here are the discriminants for the equations listed in the worksheet:
1. \( 2x^2 + 5x + 6 = 0 \): \( \Delta = -23 \)
2. \( x^2 - 4x + 4 = 0 \): \( \Delta = 0 \)
3. \( 7x^2 + 6 = 0 \): \( \Delta = -168 \)
4. \( x^2 - 3x + 4 = 0 \): \( \Delta = -7 \)
5. \( 3x^2 + 6x + 6 = 0 \): \( \Delta = -36 \)
6. \( 9y^2 - 8y - 1 = 0 \): \( \Delta = 100 \)
7. \( 5x^2 = 0 \): \( \Delta = 0 \)
8. \( 12x^2 + 2x = 0 \): \( \Delta = 4 \)
9. \( 6d^2 - 5 = 0 \): \( \Delta = 120 \)

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Boxed Final Answer

\[
\boxed{-23, 0, -168, -7, -36, 100, 0, 4, 120}
\]