The sum of two numbers is 25 and sum of their square is 313. Calculate the numbers.

Options

A. 15, 10
B. 18, 7
C. 11, 14
D. 12, 13

Answer

To solve the problem, we need to find two numbers \( x \) and \( y \) such that:

1. The sum of the two numbers is 25:
\[ x + y = 25 \]

2. The sum of their squares is 313:
\[ x^2 + y^2 = 313 \]

### Step 1: Express \( y \) in terms of \( x \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 25 - x \]

**Hint:** Use the first equation to express one variable in terms of the other.

### Step 2: Substitute \( y \) in the second equation
Now we substitute \( y \) into the second equation:
\[ x^2 + (25 - x)^2 = 313 \]

### Step 3: Expand the equation
Expanding \( (25 - x)^2 \):
\[ x^2 + (625 - 50x + x^2) = 313 \]
Combining like terms:
\[ 2x^2 - 50x + 625 = 313 \]

### Step 4: Simplify the equation
Now, we simplify the equation:
\[ 2x^2 - 50x + 625 - 313 = 0 \]
\[ 2x^2 - 50x + 312 = 0 \]

### Step 5: Factor the quadratic equation
Next, we can factor the quadratic equation:
\[ 2(x^2 - 25x + 156) = 0 \]
To factor \( x^2 - 25x + 156 \), we look for two numbers that multiply to 156 and add to -25. The factors are -13 and -12:
\[ (x - 13)(x - 12) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 13 = 0 \quad \Rightarrow \quad x = 13 \]
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]

### Step 7: Find corresponding values of \( y \)
Now, we can find the corresponding values of \( y \):
1. If \( x = 13 \):
\[ y = 25 - 13 = 12 \]

2. If \( x = 12 \):
\[ y = 25 - 12 = 13 \]

### Conclusion
The two numbers are \( 12 \) and \( 13 \).

**Final Answer:** The numbers are \( 12 \) and \( 13 \).

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