If the product of two numbers is 10 and their sum is 7, then the larger of the two numbers is

Options

A. `-2`
B. 2
C. 5
D. 3

Explanation

To solve the problem step by step, we need to find two numbers whose product is 10 and whose sum is 7. Let's denote the two numbers as \( A \) and \( B \).

### Step 1: Set up the equations
We know from the problem statement:
1. \( A \times B = 10 \) (Equation 1)
2. \( A + B = 7 \) (Equation 2)

### Step 2: Express one variable in terms of the other
From Equation 2, we can express \( A \) in terms of \( B \):
\[ A = 7 - B \]

### Step 3: Substitute into the product equation
Now, substitute \( A \) in Equation 1:
\[ (7 - B) \times B = 10 \]
Expanding this gives:
\[ 7B - B^2 = 10 \]

### Step 4: Rearrange the equation
Rearranging the equation to standard quadratic form:
\[ B^2 - 7B + 10 = 0 \]

### Step 5: Factor the quadratic equation
Next, we factor the quadratic equation:
\[ (B - 5)(B - 2) = 0 \]
This gives us two possible solutions for \( B \):
\[ B = 5 \quad \text{or} \quad B = 2 \]

### Step 6: Find the corresponding values of A
Now, we can find the corresponding values of \( A \) for both cases:
1. If \( B = 5 \):
\[ A = 7 - 5 = 2 \]
2. If \( B = 2 \):
\[ A = 7 - 2 = 5 \]

### Step 7: Identify the larger number
The two numbers are \( 2 \) and \( 5 \). The larger of the two numbers is:
\[ \text{Larger number} = 5 \]

### Final Answer
Thus, the larger of the two numbers is \( 5 \).

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