The sum of the squares of two consecutive integers is 85. What are the integers?
Understand the Problem
The question is asking to find two consecutive integers whose squares add up to 85. This involves setting up an equation and solving for the integers.
Answer
The consecutive integers are $6$ and $7$, or $-7$ and $-6$.
Answer for screen readers
The consecutive integers are $6$ and $7$, or $-7$ and $-6$.
Steps to Solve
- Define the consecutive integers
Let the first integer be $x$. Then the next consecutive integer can be defined as $x + 1$.
- Set up the equation
Write the equation using the information provided: The sum of the squares of these integers is 85. Therefore, we can set up the equation: $$x^2 + (x + 1)^2 = 85$$
- Expand the equation
Now expand the equation: $$x^2 + (x^2 + 2x + 1) = 85$$ Combine like terms: $$2x^2 + 2x + 1 = 85$$
- Simplify the equation
Rearrange the equation to set it to 0: $$2x^2 + 2x + 1 - 85 = 0$$ This simplifies to: $$2x^2 + 2x - 84 = 0$$
- Divide the equation by 2
To make calculations easier, divide the entire equation by 2: $$x^2 + x - 42 = 0$$
- Factor the quadratic
Now, factor the quadratic equation: $$(x + 7)(x - 6) = 0$$
- Find the solutions
Set each factor to zero:
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$x + 7 = 0 \Rightarrow x = -7$
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$x - 6 = 0 \Rightarrow x = 6$
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Identify the consecutive integers
For $x = 6$, the integers are $6$ and $7$. For $x = -7$, the integers are $-7$ and $-6$.
The consecutive integers are $6$ and $7$, or $-7$ and $-6$.
More Information
The integers $6$ and $7$ are two positive consecutive integers whose squares add up to $85$. The same goes for the negative integers $-7$ and $-6$. This illustrates that both positive and negative consecutive integers can satisfy the condition.
Tips
- Forgetting to square both integers when setting up the equation.
- Failing to simplify or miscalculating during factoring.
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