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Solving the Equation: (x+4)(x+1) - 3√(x^2+5x+2) = 6
In this article, we will solve the equation (x+4)(x+1) - 3√(x^2+5x+2) = 6. This equation involves a combination of algebraic and transcendental functions.
Step 1: Expand the Product
First, let's expand the product (x+4)(x+1) using the distributive property:
(x+4)(x+1) = x^2 + 5x + 4
So, the equation becomes:
x^2 + 5x + 4 - 3√(x^2+5x+2) = 6
Step 2: Isolate the Radical
Next, let's isolate the radical term -3√(x^2+5x+2) by adding it to both sides of the equation:
x^2 + 5x + 4 = 6 + 3√(x^2+5x+2)
Step 3: Square Both Sides
To eliminate the radical, we'll square both sides of the equation:
(x^2 + 5x + 4)^2 = (6 + 3√(x^2+5x+2))^2
Step 4: Expand and Simplify
Now, let's expand and simplify both sides of the equation:
x^4 + 10x^3 + 37x^2 + 60x + 16 = 36 + 36√(x^2+5x+2) + 108x^2 + 540x + 1080
Step 5: Equate Coefficients
By equating coefficients of like terms, we can set up a system of equations:
x^4 + 10x^3 + 29x^2 + 60x - 64 = 0
This is a quartic equation, which can be solved using numerical methods or algebraic manipulations.
Conclusion
In this article, we've shown the steps to solve the equation (x+4)(x+1) - 3√(x^2+5x+2) = 6. The solution involves expanding the product, isolating the radical, squaring both sides, expanding and simplifying, and equating coefficients. The final quartic equation can be solved using numerical or algebraic methods.
