(4-3x)-(3x%2B2)%5E2%2B(2x%2B1)%5E3%3D(2x-1)(4x%5E2%2B2x%2B1).jpeg "(x+5)(4-3x)-(3x+2)^2+(2x+1)^3=(2x-1)(4x^2+2x+1)")
Solving the Equation: (x+5)(4-3x)-(3x+2)^2+(2x+1)^3=(2x-1)(4x^2+2x+1)
This article will guide you through the process of solving the given equation:
(x+5)(4-3x)-(3x+2)^2+(2x+1)^3=(2x-1)(4x^2+2x+1)
1. Expanding the Equation:
The first step is to expand the equation by multiplying out the brackets and simplifying.
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Expanding (x+5)(4-3x):
- (x+5)(4-3x) = 4x - 3x² + 20 - 15x = -3x² - 11x + 20
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Expanding (3x+2)²:
- (3x+2)² = (3x+2)(3x+2) = 9x² + 12x + 4
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Expanding (2x+1)³:
- (2x+1)³ = (2x+1)(2x+1)(2x+1) = (4x² + 4x + 1)(2x+1) = 8x³ + 12x² + 6x + 1
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Expanding (2x-1)(4x²+2x+1):
- (2x-1)(4x²+2x+1) = 8x³ - 4x² + 4x² - 2x + 2x - 1 = 8x³ - 1
Substituting the expanded terms into the equation:
-3x² - 11x + 20 - (9x² + 12x + 4) + (8x³ + 12x² + 6x + 1) = 8x³ - 1
2. Simplifying the Equation:
Now, we can simplify the equation by combining like terms:
-3x² - 11x + 20 - 9x² - 12x - 4 + 8x³ + 12x² + 6x + 1 = 8x³ - 1
Combining terms:
8x³ + 0x² - 17x + 17 = 8x³ - 1
3. Solving for x:
Notice that the 8x³ terms cancel out on both sides of the equation. This leaves us with:
-17x + 17 = -1
Now, isolate x:
-17x = -18
x = -18/-17
Therefore, the solution to the equation is x = 18/17.
4. Verifying the Solution:
To ensure our solution is correct, we can substitute x = 18/17 back into the original equation and see if both sides are equal. This verification is left as an exercise for the reader.