The sum of two numbers is 9 and their product is 20. Find the sum of their:
cubes

Answer

To solve the problem step by step, we need to find two numbers whose sum is 9 and product is 20, and then calculate the sum of their cubes.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we have the following two equations:
1. \( x + y = 9 \) (Equation 1)
2. \( xy = 20 \) (Equation 2)

### Step 3: Express One Variable in Terms of the Other
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 9 - y \]

### Step 4: Substitute into the Second Equation
Now, substitute \( x \) in Equation 2:
\[ (9 - y)y = 20 \]
Expanding this gives:
\[ 9y - y^2 = 20 \]

### Step 5: Rearrange the Equation
Rearranging the equation gives us:
\[ y^2 - 9y + 20 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we need to factor the quadratic equation:
\[ (y - 4)(y - 5) = 0 \]
This gives us two possible solutions for \( y \):
1. \( y = 4 \)
2. \( y = 5 \)

### Step 7: Find Corresponding Values of \( x \)
Using the values of \( y \) to find \( x \):
- If \( y = 4 \), then \( x = 9 - 4 = 5 \).
- If \( y = 5 \), then \( x = 9 - 5 = 4 \).

Thus, the two numbers are \( 4 \) and \( 5 \).

### Step 8: Calculate the Sum of Their Cubes
Now we need to find the sum of their cubes:
\[ x^3 + y^3 = 4^3 + 5^3 \]
Calculating each cube:
\[ 4^3 = 64 \quad \text{and} \quad 5^3 = 125 \]
So,
\[ x^3 + y^3 = 64 + 125 = 189 \]

### Final Answer
The sum of the cubes of the two numbers is \( 189 \).

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