Question

Find two consecutive negative integers, sum of whose squares is 481.

Hint:

Always verify your answers:
\[ (-16)^2 + (-15)^2 = 256 + 225 = 481 \] This confirms the solution is correct!

Answer 1

Step 1: Understanding the Question:
This question is a word problem that translates into a quadratic equation.
We need to find two consecutive negative integers such that the sum of their squares is $481$.
Consecutive integers are numbers that follow each other in order, with a difference of $1$.

Step 2: Key Formula or Approach:
1. Let the first negative integer be $x$.
2. Since the integers are consecutive, the second negative integer is $x + 1$.
3. Formulate the equation based on the given condition:
\[ x^2 + (x + 1)^2 = 481 \] 4. Expand the equation, collect terms to form a standard quadratic equation $ax^2 + bx + c = 0$, and solve for $x$ using factorization or the quadratic formula.
5. Filter out any positive solutions since the question specifies "negative integers".

Step 3: Detailed Explanation:

• Set up the algebraic equation:
\[ x^2 + (x + 1)^2 = 481 \]

• Expand the term $(x+1)^2$:
\[ x^2 + (x^2 + 2x + 1) = 481 \] \[ 2x^2 + 2x + 1 = 481 \]

• Subtract 481 from both sides to set the quadratic to zero:
\[ 2x^2 + 2x + 1 - 481 = 0 \] \[ 2x^2 + 2x - 480 = 0 \]

• Divide the entire equation by 2 to simplify:
\[ x^2 + x - 240 = 0 \]

• Solve the quadratic equation by splitting the middle term:
We need two numbers that multiply to $-240$ and add up to $+1$.
Let us test factors of 240:
\[ 15 \times 16 = 240 \] Therefore, the numbers are $+16$ and $-15$.
Rewrite the middle term:
\[ x^2 + 16x - 15x - 240 = 0 \]

• Factor by grouping:
\[ x(x + 16) - 15(x + 16) = 0 \] \[ (x - 15)(x + 16) = 0 \]

• This gives two mathematical solutions:
\[ x = 15 \quad \text{or} \quad x = -16 \]

• Apply the "negative integer" constraint:
Since $x$ must be a negative integer, we reject $x = 15$.
Therefore:
\[ x = -16 \]

• Find the second consecutive integer:
\[ x + 1 = -16 + 1 = -15 \] Both $-16$ and $-15$ are negative integers.


Step 4: Final Answer:
The two consecutive negative integers are $-16$ and $-15$.