2-16%3D0.jpeg "(x-7)2-16=0")
Solving the Quadratic Equation: (x-7)² - 16 = 0
This article will guide you through solving the quadratic equation (x-7)² - 16 = 0. We'll utilize the concepts of factoring and square roots to find the solutions for x.
Understanding the Equation
The given equation is a quadratic equation because it contains a term with x². It's also in a simplified form where we can directly apply factoring techniques.
Solving by Factoring
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Recognize the Difference of Squares: Notice that the equation is in the form of a² - b² = 0, where a = (x-7) and b = 4. We can factor this using the difference of squares pattern: (a+b)(a-b) = 0.
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Factor the Equation: Applying the pattern, we get: [(x-7) + 4] [(x-7) - 4] = 0 (x - 3)(x - 11) = 0
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Solve for x: For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:
- x - 3 = 0 => x = 3
- x - 11 = 0 => x = 11
Solving by Square Roots
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Isolate the Squared Term: Start by adding 16 to both sides of the equation: (x - 7)² = 16
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Take the Square Root of Both Sides: Remember that when taking the square root, we need to consider both positive and negative possibilities: √(x - 7)² = ±√16
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Simplify: This gives us: x - 7 = ±4
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Solve for x: Now, isolate x by adding 7 to both sides: x = 7 ± 4
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Final Solutions: Therefore, we have two solutions:
- x = 7 + 4 = 11
- x = 7 - 4 = 3
Conclusion
Both factoring and square root methods lead to the same solutions for the equation (x-7)² - 16 = 0: x = 3 and x = 11. These are the two values of x that satisfy the original equation.