(x%2B2)(x%2B3)(x%2B4)%3D840.jpeg "(x+1)(x+2)(x+3)(x+4)=840")
Solving the Equation: (x+1)(x+2)(x+3)(x+4)=840
In this article, we will explore the solution to the equation (x+1)(x+2)(x+3)(x+4)=840. This equation is a product of four binomials, and solving it will require some algebraic manipulations and clever thinking.
Expanding the Equation
The first step in solving this equation is to expand the product of the four binomials. This can be done using the distributive property of multiplication over addition. After expanding the equation, we get:
x^4 + 10x^3 + 35x^2 + 50x + 24 = 840
Simplifying the Equation
Next, we can simplify the equation by subtracting 840 from both sides:
x^4 + 10x^3 + 35x^2 + 50x - 816 = 0
This is a quartic equation, which can be challenging to solve. However, we can try to find a solution by factoring or using numerical methods.
Factoring the Equation
After some trial and error, we find that:
x^4 + 10x^3 + 35x^2 + 50x - 816 = (x + 8)(x + 4)(x - 3)(x - 5) = 0
This factorization suggests that there are four possible values of x that satisfy the equation:
x + 8 = 0 => x = -8 x + 4 = 0 => x = -4 x - 3 = 0 => x = 3 x - 5 = 0 => x = 5
Conclusion
In conclusion, the equation (x+1)(x+2)(x+3)(x+4)=840 has four possible solutions: x = -8, x = -4, x = 3, and x = 5. These solutions can be obtained by expanding the equation, simplifying it, and factoring it.
