Solve the following equation:
x+2(2x+1)2+2x+1(x+2)2=4.5x
In order to solve the equation, start by removing the denominators.
To do this, we'll multiply the denominators:
(2x+1)2⋅(2x+1)+(x+2)2⋅(x+2)=4.5x(2x+1)(x+2)
Open the parentheses on the left side, making use of the distributive property:
(4x2+4x+1)⋅(2x+1)+(x2+4x+4)⋅(x+2)=4.5x(2x+1)(x+2)
Continue to open the parentheses on the right side of the equation:
(4x2+4x+1)⋅(2x+1)+(x2+4x+4)⋅(x+2)=4.5x(2x2+5x+2)
Simplify further:
(4x2+4x+1)⋅(2x+1)+(x2+4x+4)⋅(x+2)=9x3+22.5x+9x
Go back and simplify the parentheses on the left side of the equation:
8x3+8x2+2x+4x2+4x+1+x3+4x2+4x+2x2+8x+8=9x3+22.5x+9x
Combine like terms:
9x3+18x2+18x+9=9x3+22.5x+9x
Notice that all terms can be divided by 9 as shown below:
x3+2x2+2x+1=x3+2.5x+x
Move all numbers to one side:
x3−x3+2x2−2.5x2+2x−x+9=0
We obtain the following:
0.5x2−x−1=0
In order to remove the one-half coefficient, multiply the entire equation by 2
x2−2x−2=0
Apply the square root formula, as shown below-
22±12
Apply the properties of square roots in order to simplify the square root of 12:
22±23Divide both the numerator and denominator by 2 as follows:
1±3
x=1±3