(x-3)(x-5)(x-7)%3D9.jpeg "(x-1)(x-3)(x-5)(x-7)=9")
Solving the Equation (x-1)(x-3)(x-5)(x-7) = 9
This equation presents a unique challenge due to the presence of four factors on the left-hand side. Let's explore how to find its solutions.
Understanding the Problem
The equation (x-1)(x-3)(x-5)(x-7) = 9 involves finding values of x that, when plugged into the equation, make it true. This means we need to find the roots of the polynomial on the left side of the equation.
Solving Strategies
1. Expanding the Equation
One approach is to expand the left-hand side of the equation by multiplying out the factors:
(x-1)(x-3)(x-5)(x-7) = (x^2 - 4x + 3)(x^2 - 12x + 35) = x^4 - 16x^3 + 77x^2 - 168x + 105
This gives us a fourth-degree polynomial equation: x^4 - 16x^3 + 77x^2 - 168x + 96 = 0
Solving fourth-degree polynomials can be quite challenging, and there's no general formula like the quadratic formula.
2. Using Numerical Methods
We can use numerical methods like the Newton-Raphson method or bisection method to approximate the solutions. These methods involve iterative calculations and can provide very precise solutions.
3. Recognizing Patterns
Observing the pattern in the factors, we notice that the constant terms in each factor are consecutive odd numbers (1, 3, 5, 7). This might suggest a potential approach:
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Guessing: We can try plugging in some values for x to see if any lead to a solution. For example, if we try x = 2, the left side becomes (2-1)(2-3)(2-5)(2-7) = 1 * -1 * -3 * -5 = 15. This is not 9, but it shows that certain values can lead to relatively close results.
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Graphical Analysis: By graphing the function y = (x-1)(x-3)(x-5)(x-7) - 9, we can visually identify the points where the graph intersects the x-axis. These points represent the solutions of the equation.
Conclusion
The equation (x-1)(x-3)(x-5)(x-7) = 9 is not easily solved by direct algebraic manipulation. However, by employing numerical methods, graphical analysis, or even careful guesswork, we can approximate the solutions or identify them visually.