https://file.aiquickdraw.com/mathbot/file/1726065886713aak9ials.jpg

We are given two rational numbers \( a \) and \( b \) that satisfy the following conditions:

\[
ab = \frac{-5}{3} \quad \text{and} \quad a + b = \frac{-7}{9}
\]

We need to simplify the following expressions:

1. \( A = (a + 1)(b + 1) \)
2. \( B = (a - 3)(b - 3) \)
3. \( C = \frac{a}{b} + \frac{b}{a} + 2 \)

### Simplifying Expression \( A = (a + 1)(b + 1) \)

Expanding this expression:

\[
A = (a + 1)(b + 1) = ab + a + b + 1
\]

Using the given values:

\[
ab = \frac{-5}{3} \quad \text{and} \quad a + b = \frac{-7}{9}
\]

Substitute these values into the expression for \( A \):

\[
A = \frac{-5}{3} + \frac{-7}{9} + 1
\]

To simplify, first bring everything to a common denominator. The common denominator between 3, 9, and 1 is 9:

\[
A = \frac{-15}{9} + \frac{-7}{9} + \frac{9}{9}
\]

\[
A = \frac{-15 - 7 + 9}{9} = \frac{-13}{9}
\]

Thus, \( A = \frac{-13}{9} \).

---

### Simplifying Expression \( B = (a - 3)(b - 3) \)

Expanding this expression:

\[
B = (a - 3)(b - 3) = ab - 3a - 3b + 9
\]

Using the given values \( ab = \frac{-5}{3} \) and \( a + b = \frac{-7}{9} \), we substitute:

\[
B = \frac{-5}{3} - 3(a + b) + 9
\]

Substitute \( a + b = \frac{-7}{9} \):

\[
B = \frac{-5}{3} - 3 \times \frac{-7}{9} + 9
\]

Simplify:

\[
B = \frac{-5}{3} + \frac{7}{3} + 9
\]

The first two terms can be combined:

\[
B = \frac{-5 + 7}{3} + 9 = \frac{2}{3} + 9
\]

Convert 9 to have a denominator of 3:

\[
B = \frac{2}{3} + \frac{27}{3} = \frac{29}{3}
\]

Thus, \( B = \frac{29}{3} \).

---

### Simplifying Expression \( C = \frac{a}{b} + \frac{b}{a} + 2 \)

The expression can be simplified using the identity \( \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} \):

\[
C = \frac{a^2 + b^2}{ab} + 2
\]

To find \( a^2 + b^2 \), we use the identity:

\[
a^2 + b^2 = (a + b)^2 - 2ab
\]

Substitute the given values \( a + b = \frac{-7}{9} \) and \( ab = \frac{-5}{3} \):

\[
a^2 + b^2 = \left( \frac{-7}{9} \right)^2 - 2 \times \frac{-5}{3}
\]

\[
a^2 + b^2 = \frac{49}{81} + \frac{10}{3}
\]

Convert \( \frac{10}{3} \) to have a denominator of 81:

\[
a^2 + b^2 = \frac{49}{81} + \frac{270}{81} = \frac{319}{81}
\]

Now, substitute back into the expression for \( C \):

\[
C = \frac{\frac{319}{81}}{\frac{-5}{3}} + 2 = \frac{319}{81} \times \frac{3}{-5} + 2
\]

\[
C = \frac{319 \times 3}{81 \times -5} + 2 = \frac{957}{-405} + 2
\]

Simplify the fraction:

\[
C = \frac{-319}{135} + 2
\]

Convert 2 to have a denominator of 135:

\[
C = \frac{-319}{135} + \frac{270}{135} = \frac{-319 + 270}{135} = \frac{-49}{135}
\]

Thus, \( C = \frac{-49}{135} \).

---

### Summary of Results:
1. \( A = \frac{-13}{9} \)
2. \( B = \frac{29}{3} \)
3. \( C = \frac{-49}{135} \)

Would you like any more details or have any further questions? Here are some related questions to explore:

1. How can we solve for \( a \) and \( b \) given their sum and product?
2. What is the geometric interpretation of the given conditions \( ab = \frac{-5}{3} \) and \( a + b = \frac{-7}{9} \)?
3. Can the expression \( C \) be simplified using another algebraic approach?
4. How would these calculations change if \( ab \) and \( a + b \) were integers instead of fractions?
5. What are other ways to simplify expressions like \( A = (a+1)(b+1) \) or \( B = (a-3)(b-3) \)?

**Tip:** When simplifying algebraic expressions, using identities and factoring can often make the process more efficient.

Soit a et b deux nombres rationnels tels que : ab = -5/3 et a + b = -7/9. Simplifier les expressions suivantes : A = (a+1)(b+1), B = (a-3)(b-3), C = a/b + b/a + 2

Learn how to simplify algebraic expressions involving two rational numbers a and b, where ab = -5/3 and a + b = -7/9. This problem involves core algebraic concepts such as rational number operations and algebraic identities. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation