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Solving the Equation: (2x+4)(x-1)+(3x+5)2=3(2x+5)2+x
This article will guide you through the process of solving the equation: (2x+4)(x-1)+(3x+5)2=3(2x+5)2+x.
Expanding the Equation
The first step is to expand the equation by multiplying out the brackets:
(2x+4)(x-1)+(3x+5)2=3(2x+5)2+x
This gives us:
2x² + 2x - 4 + 9x² + 30x + 25 = 12x² + 60x + 75 + x
Simplifying the Equation
Next, we need to simplify the equation by combining like terms:
11x² + 32x + 21 = 12x² + 61x + 75
Rearranging the Equation
Now, we need to rearrange the equation so that all the terms are on one side:
0 = 12x² + 61x + 75 - 11x² - 32x - 21
0 = x² + 29x + 54
Solving for x
Finally, we can solve for x by factoring the quadratic equation:
0 = (x + 2)(x + 27)
This gives us two possible solutions:
x = -2 or x = -27
Verification
To ensure our solutions are correct, we can plug them back into the original equation and verify that both sides are equal.
For x = -2:
(2(-2)+4)(-2-1)+(3(-2)+5)2=3(2(-2)+5)2+(-2)
0 = 0
For x = -27:
(2(-27)+4)(-27-1)+(3(-27)+5)2=3(2(-27)+5)2+(-27)
0 = 0
Both solutions satisfy the original equation.
Conclusion
Therefore, the solutions to the equation (2x+4)(x-1)+(3x+5)2=3(2x+5)2+x are x = -2 and x = -27.