find the sum and the product of the roots of the quadratic equation 1) m²-2m-6=0 2) 2j²-7j-3=0 3) m²-3m=2

To find the sum and product of the roots of quadratic equations, we can use the relationships derived from **Vieta's formulas**. For any quadratic equation of the form:

\[
ax^2 + bx + c = 0
\]

- The **sum of the roots** is given by: 
  \[
  \text{Sum of roots} = -\frac{b}{a}
  \]
- The **product of the roots** is given by: 
  \[
  \text{Product of roots} = \frac{c}{a}
  \]

Let's apply these formulas to each equation:

### 1) \( m^2 - 2m - 6 = 0 \)

Here, \( a = 1 \), \( b = -2 \), and \( c = -6 \).

- **Sum of roots**:
  \[
  \text{Sum} = -\frac{-2}{1} = 2
  \]
- **Product of roots**:
  \[
  \text{Product} = \frac{-6}{1} = -6
  \]

### 2) \( 2j^2 - 7j - 3 = 0 \)

Here, \( a = 2 \), \( b = -7 \), and \( c = -3 \).

- **Sum of roots**:
  \[
  \text{Sum} = -\frac{-7}{2} = \frac{7}{2}
  \]
- **Product of roots**:
  \[
  \text{Product} = \frac{-3}{2} = -\frac{3}{2}
  \]

### 3) \( m^2 - 3m = 2 \)

First, rearrange the equation to standard form:
\[
m^2 - 3m - 2 = 0
\]
Here, \( a = 1 \), \( b = -3 \), and \( c = -2 \).

- **Sum of roots**:
  \[
  \text{Sum} = -\frac{-3}{1} = 3
  \]
- **Product of roots**:
  \[
  \text{Product} = \frac{-2}{1} = -2
  \]

### Summary:
1. \( m^2 - 2m - 6 = 0 \): Sum = 2, Product = -6
2. \( 2j^2 - 7j - 3 = 0 \): Sum = \( \frac{7}{2} \), Product = \( -\frac{3}{2} \)
3. \( m^2 - 3m - 2 = 0 \): Sum = 3, Product = -2

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Would you like any additional details or have any further questions?

Here are 5 related questions:
1. How do you solve for the actual roots of these quadratic equations?
2. What is the significance of Vieta's formulas in algebra?
3. How can these relationships be applied to complex roots?
4. Can the sum and product of roots be used to verify factored forms of quadratics?
5. How can we generalize this for cubic equations?

**Tip:** To factor a quadratic equation, first try to express the middle term as a sum or difference that matches the product of the first and last coefficients!

Learn how to find the sum and product of the roots for quadratic equations like m²-2m-6=0 and 2j²-7j-3=0 using Vieta's formulas. This solution breaks down the algebraic concepts and formulas step-by-step, making it suitable for students in Grades 8-10.

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