https://r2.erweima.ai/mathbot/image/17267081708024piq9tje.jpg

The equation provided in the image is:

\[
2x^2 + 10x + 7 = 0
\]

The question asks for the **sum** and **product** of the two roots of this quadratic equation.

For a quadratic equation in the standard form:

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

The sum and product of the roots can be found using the following formulas:

- **Sum of the roots** \( S \) = \( -\frac{b}{a} \)
- **Product of the roots** \( P \) = \( \frac{c}{a} \)

Here, the coefficients are:
- \( a = 2 \)
- \( b = 10 \)
- \( c = 7 \)

Now let's calculate:

1. **Sum of the roots**:
\[
S = -\frac{b}{a} = -\frac{10}{2} = -5
\]

2. **Product of the roots**:
\[
P = \frac{c}{a} = \frac{7}{2}
\]

So the sum of the roots is \( -5 \) and the product is \( \frac{7}{2} \).

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions for you to explore:
1. How do you derive the quadratic formula for solving quadratic equations?
2. What are the conditions for real, imaginary, or equal roots in a quadratic equation?
3. Can the discriminant be used to predict the nature of the roots?
4. What are the practical applications of solving quadratic equations?
5. How can we factor quadratic equations when roots are integers?

**Tip**: For any quadratic equation, the discriminant \( \Delta = b^2 - 4ac \) helps to determine the nature of the roots (real and distinct, real and equal, or complex).

Find the sum and product of the two solutions for the equation 2x^2 + 10x + 7 = 0.

Learn how to calculate the sum and product of the roots for the quadratic equation 2x^2 + 10x + 7. This explanation covers algebraic concepts such as the sum and product of roots theorem, and is suitable for students in Grades 8-10.

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