How do you solve the equation x² + 18x + 81 = 25?
Introduction
Have you ever looked at an algebraic equation and felt like you were trying to translate a foreign language? You are not alone! Mathematics, much like English, has its own syntax, grammar, and structure.
Today, we are going to translate and solve a classic quadratic equation: x² + 18x + 81 = 25.
By breaking this down step-by-step, we will see how the "grammar" of algebra allows us to simplify and solve this elegant problem using two different methods: Factoring a Perfect Square Trinomial and the Standard Quadratic Method.
Definitions
Before we dive into the calculations, let's define our key mathematical terms so we are all speaking the same language:
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Quadratic Equation: An equation of the second degree, meaning it contains at least one term that is squared (raised to the power of 2, like x²).
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Perfect Square Trinomial: A three-term algebraic expression that can be factored into the square of a binomial (e.g., a² + 2ab + b² = (a + b)²).
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Binomial: An algebraic expression consisting of two terms (e.g., x + 9).
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Square Root: A value that, when multiplied by itself, gives the original number (e.g., the square root of 25 is ±5).
Quick Reference Table
Here is a quick summary of the two primary methods we can use to solve this equation:
| Method | Core Concept | Best Used For | Difficulty Level |
|---|---|---|---|
| Perfect Square Method | Recognizing that x² + 18x + 81 is (x + 9)² and taking the square root of both sides. | Equations where the left side is already a perfect square. | Easy / Elegant |
| Standard Quadratic Method | Setting the equation to zero (ax² + bx + c = 0) and factoring or using the quadratic formula. | Any quadratic equation, especially when factoring isn't obvious. | Moderate / Universal |
Step-by-Step Guide: Method 1 (The Perfect Square Method)
This is the most elegant and direct way to solve this specific equation because the left side is a perfect square trinomial.
Step 1: Recognize the Pattern
Look closely at the left side of the equation:
x² + 18x + 81 = 25
Notice that:
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The first term, x², is the square of x.
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The last term, 81, is the square of 9 (9 × 9 = 81).
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The middle term, 18x, is exactly twice the product of x and 9 (2 × x × 9 = 18x).
This fits the perfect square trinomial formula perfectly: a² + 2ab + b² = (a + b)².
Step 2: Rewrite as a Binomial Squared
Since we recognized the pattern, we can rewrite the left side of the equation:
(x + 9)² = 25
Step 3: Take the Square Root of Both Sides
To undo the square on the left side, we take the square root of both sides of the equation. Remember, a positive number has both a positive and a negative square root!
√(x + 9)² = ±√25
x + 9 = ±5
Step 4: Solve the Two Resulting Equations
This gives us two separate linear equations to solve:
Case 1 (Positive):
x + 9 = 5
x = 5 - 9
x = -4
Case 2 (Negative):
x + 9 = -5
x = -5 - 9
x = -14
Our solutions are x = -4 and x = -14.
Step-by-Step Guide: Method 2 (The Standard Quadratic Method)
If you didn't notice the perfect square right away, don't worry! You can always solve a quadratic equation by setting it to zero first.
Step 1: Set the Equation to Zero
Subtract 25 from both sides to get the equation into standard form (ax² + bx + c = 0):
x² + 18x + 81 - 25 = 0
x² + 18x + 56 = 0
Step 2: Factor the Trinomial
Now, we need to find two numbers that multiply to 56 and add up to 18.
Let's think of the factors of 56:
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1 × 56 (adds to 57)
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2 × 28 (adds to 30)
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4 × 14 (adds to 18) — Bingo!
So, we can factor the equation as:
(x + 4)(x + 14) = 0
Step 3: Apply the Zero Product Property
If two factors multiply to equal zero, at least one of them must be zero:
x + 4 = 0 => x = -4
x + 14 = 0 => x = -14
Both methods yield the exact same beautiful results!
Common Pitfalls to Avoid
When solving quadratic equations, language and math students alike often make these common slips:
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Forgetting the ± Sign: When taking the square root of 25, it is easy to write just 5. Remember that (-5) × (-5) is also 25. Forgetting this means you will lose half of your solution!
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Sign Errors: When moving terms across the equals sign (e.g., subtracting 9 from both sides), pay close attention to whether your final answer should be positive or negative.
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Misidentifying the Perfect Square: Always verify that the middle term is indeed 2ab before factoring it as (a+b)².
Real-World Analogy
Think of solving a quadratic equation like solving a mystery in a novel. The clues are all there in the text (the equation). You can either use a shortcut because you recognize the culprit's signature style (Method 1: Perfect Squares), or you can systematically interview every suspect until you find the truth (Method 2: Standard Factoring). Both paths lead to justice!