(x+4)(x+5)(x+7)(x+8)=4

Jasonbradley

3 min read

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Jun 17, 2024

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Solving the Equation (x+4)(x+5)(x+7)(x+8) = 4

This equation presents a unique challenge because it's a quartic equation (an equation with the highest power of x being 4). We can't easily factor it like a quadratic. However, there are methods to approach solving it:

1. Expanding and Simplifying:

• Expand the product: Multiply out the terms on the left side of the equation. This will result in a polynomial with x terms up to the fourth power.
• Rearrange and set equal to zero: Move all the terms to one side to get a standard form of the quartic equation.
• Attempt to factor: See if you can factor the quartic equation. This might require some clever algebraic manipulation.

2. Using the Rational Root Theorem:

• The Rational Root Theorem helps us find potential rational roots (roots that can be expressed as fractions).
• Identify factors of the constant term and leading coefficient: In our equation, the constant term is 4 and the leading coefficient is 1.
• Create possible fractions: Form fractions using the factors of the constant term (1, 2, 4) as numerators and the factors of the leading coefficient (1) as denominators. This gives us potential rational roots: ±1, ±2, ±4.
• Test the potential roots: Substitute each potential root into the equation and check if it results in zero. If it does, that value is a root.

3. Numerical Methods:

• Newton-Raphson Method: This iterative method uses calculus to approximate roots of an equation.
• Graphical Method: Plot the graph of the function (left side of the equation minus the right side). The x-intercepts of the graph represent the roots.

4. Using a Computer Algebra System:

• Software like Wolfram Alpha or Mathematica can easily solve such equations. They provide both numerical approximations and exact solutions (if possible).

Important Note: While these methods provide approaches, finding exact solutions to quartic equations can be challenging and might involve complex numbers. In many practical cases, numerical approximations are sufficient.