For Mathematics:
(1/2 cross wise pad paper) 
Write the quadratic equation of the following:
1. 8 and 2
2. - 3 and 7
3. 2/3 and - 6
4. - 3 and - 5

To write the quadratic equation given the roots of the equation, we use the fact that if \( r_1 \) and \( r_2 \) are the roots, the quadratic equation can be written as:

\[
(x - r_1)(x - r_2) = 0
\]

Expanding this, the quadratic equation becomes:

\[
x^2 - (r_1 + r_2)x + r_1r_2 = 0
\]

Now, let's apply this to each of the given pairs of roots.

### 1. Roots: 8 and 2
- Sum of roots \( r_1 + r_2 = 8 + 2 = 10 \)
- Product of roots \( r_1r_2 = 8 \times 2 = 16 \)

The quadratic equation is:

\[
x^2 - 10x + 16 = 0
\]

### 2. Roots: -3 and 7
- Sum of roots \( r_1 + r_2 = -3 + 7 = 4 \)
- Product of roots \( r_1r_2 = -3 \times 7 = -21 \)

The quadratic equation is:

\[
x^2 - 4x - 21 = 0
\]

### 3. Roots: \(\frac{2}{3}\) and \(-6\)
- Sum of roots \( r_1 + r_2 = \frac{2}{3} + (-6) = \frac{2}{3} - \frac{18}{3} = \frac{-16}{3} \)
- Product of roots \( r_1r_2 = \frac{2}{3} \times (-6) = -4 \)

The quadratic equation is:

\[
x^2 + \frac{16}{3}x - 4 = 0
\]

If you prefer the equation without fractions, multiply the entire equation by 3:

\[
3x^2 + 16x - 12 = 0
\]

### 4. Roots: -3 and -5
- Sum of roots \( r_1 + r_2 = -3 + (-5) = -8 \)
- Product of roots \( r_1r_2 = -3 \times -5 = 15 \)

The quadratic equation is:

\[
x^2 + 8x + 15 = 0
\]

Would you like any further details or clarifications?

Here are 5 related questions:
1. How can you derive the quadratic formula from a general quadratic equation?
2. What is the discriminant, and how does it relate to the nature of the roots?
3. How can you determine the axis of symmetry for a quadratic equation?
4. What is the significance of the vertex in a quadratic equation?
5. How can quadratic equations be solved using the method of completing the square?

**Tip:** Always check the sum and product of the roots when writing the quadratic equation to ensure accuracy.

Learn how to write quadratic equations from given roots with practice problems. Includes step-by-step solutions for finding quadratic equations from pairs of roots such as 8 and 2, -3 and 7, 2/3 and -6, and -3 and -5. This problem covers Vieta's formulas and is suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation