calculate the discriminant, D, determine the nature of the roots and find their sum and products I want to see the steps and calculations in markdown format, Use simple terms and numbers:

1. 2y^2 = 8y - 26 

Let's go through the process step-by-step for solving the quadratic equation \( 2y^2 = 8y - 26 \).

### Step 1: Rearrange the equation into standard quadratic form
The standard form of a quadratic equation is:

\[
ay^2 + by + c = 0
\]

So, first, we need to move all terms to one side of the equation. Start by subtracting \( 8y \) and adding \( 26 \) from both sides:

\[
2y^2 - 8y + 26 = 0
\]

Now, the equation is in standard form where:

- \( a = 2 \)
- \( b = -8 \)
- \( c = 26 \)

### Step 2: Calculate the discriminant, \( D \)
The discriminant is calculated using the formula:

\[
D = b^2 - 4ac
\]

Substitute the values of \( a \), \( b \), and \( c \):

\[
D = (-8)^2 - 4(2)(26)
\]

Now, calculate step-by-step:

\[
D = 64 - 208
\]

\[
D = -144
\]

### Step 3: Determine the nature of the roots
The discriminant helps determine the nature of the roots:

- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is one real root (a repeated root).
- If \( D < 0 \), there are two complex roots.

Since \( D = -144 \) (which is less than 0), the equation has **two complex roots**.

### Step 4: Find the sum and product of the roots
For a quadratic equation \( ay^2 + by + c = 0 \), the sum and product of the roots are given by:

- **Sum of roots**: \( -\frac{b}{a} \)
- **Product of roots**: \( \frac{c}{a} \)

#### Sum of the roots:
\[
\text{Sum of roots} = -\frac{-8}{2} = \frac{8}{2} = 4
\]

#### Product of the roots:
\[
\text{Product of roots} = \frac{26}{2} = 13
\]

### Final Answer:
- **Discriminant (D)**: \( -144 \)
- **Nature of the roots**: Two complex roots
- **Sum of the roots**: \( 4 \)
- **Product of the roots**: \( 13 \)

---

Would you like more details on any step, or have any questions?

#### Relative Questions:
1. What are complex roots, and how do they differ from real roots?
2. How can we find the exact values of complex roots from the quadratic equation?
3. What is the significance of the discriminant in real-world applications?
4. How would the nature of roots change if the discriminant were positive?
5. Can the quadratic formula still be applied when the discriminant is negative?

#### Tip:
When solving quadratic equations, always check the discriminant first, as it gives insight into the type of solutions you'll get (real or complex).

calculate the discriminant, D, determine the nature of the roots and find their sum and products I want to see the steps and calculations in markdown format, Use simple terms and numbers:

1. 2y^2 = 8y - 26

Learn how to solve the quadratic equation 2y^2 = 8y - 26 by calculating the discriminant, determining the nature of roots (complex or real), and finding the sum and product of the roots. This detailed step-by-step guide uses key algebraic formulas and is suitable for students in Grades 9-11.

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