%5E2%3D81.jpeg "(x-7)^2=81")
Solving the Equation (x-7)^2 = 81
This article will walk through the process of solving the equation (x-7)^2 = 81. We will explore both algebraic and conceptual approaches to understand the solutions.
Understanding the Equation
The equation represents a quadratic equation, where the left side is a perfect square trinomial. This means the expression can be factored into the form (x - a)^2, where 'a' is a constant.
Solving by Square Root Property
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Isolate the squared term: The equation is already in this form.
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Take the square root of both sides: Remember to include both positive and negative roots.
√(x-7)^2 = ±√81
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Simplify:
x - 7 = ±9
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Solve for x:
x = 7 ± 9
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Find the two solutions:
- x = 7 + 9 = 16
- x = 7 - 9 = -2
Therefore, the solutions to the equation (x-7)^2 = 81 are x = 16 and x = -2.
Visualizing the Solution
The equation can also be interpreted geometrically. The equation represents a parabola that is shifted 7 units to the right and opens upwards. The solutions to the equation are the x-coordinates where the parabola intersects the line y = 9 and y = -9.
Conclusion
By using the square root property, we effectively solved the quadratic equation (x-7)^2 = 81 and found two distinct solutions: x = 16 and x = -2. Understanding the process and the visual interpretation helps us gain a deeper understanding of quadratic equations and their solutions.