To solve an equation by square roots, you'll isolate the squared term, then take the square root of both sides of the equation, remembering to account for both positive and negative solutions.
Here's a breakdown of the process:
1. Isolate the Squared Term:
- Your goal is to get the term with the variable squared (e.g., x², (x + 2)²) by itself on one side of the equation. This might involve adding, subtracting, multiplying, or dividing both sides of the equation.
2. Take the Square Root of Both Sides:
- Once the squared term is isolated, take the square root of both sides of the equation. Remember that the square root of a number has both a positive and a negative solution. For example, √9 = +3 and -3, because 3² = 9 and (-3)² = 9. Therefore, you'll need to include a "±" (plus or minus) sign when taking the square root.
3. Solve for the Variable:
- After taking the square root, you'll likely have a simpler equation. Solve for the variable using basic algebraic operations (addition, subtraction, multiplication, division). This will give you the possible values for the variable.
4. Simplify (if needed):
- Simplify the square root if possible. You can also simplify the final answers if necessary.
Example 1: Simple Equation
Solve for x: x² = 16
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Isolate the squared term: The x² term is already isolated.
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Take the square root of both sides: √x² = ±√16
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Solve for x: x = ±4
So, x = 4 or x = -4
Example 2: Equation with Additional Operations
Solve for x: 2x² - 8 = 0
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Isolate the squared term:
- Add 8 to both sides: 2x² = 8
- Divide both sides by 2: x² = 4
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Take the square root of both sides: √x² = ±√4
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Solve for x: x = ±2
So, x = 2 or x = -2
Example 3: Equation with a Binomial Squared
Solve for x: (x + 3)² = 25
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Isolate the squared term: The (x + 3)² term is already isolated.
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Take the square root of both sides: √(x + 3)² = ±√25
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Solve for x: x + 3 = ±5
- x + 3 = 5 --> x = 5 - 3 --> x = 2
- x + 3 = -5 --> x = -5 - 3 --> x = -8
So, x = 2 or x = -8
In summary, solving by square roots involves isolating the squared term, taking the square root of both sides (remembering the ±), and then solving for the variable. This method is particularly useful when the variable appears only in a squared term (or a term that can be easily transformed into a squared term).
