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`3(2^x)+2=2^(2x+1)` Solve the equation.Please state your workings clearly.Thanks
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`3(2^x) +2= 2^(2x+1)`
First, rewrite the right-hand side using the rules of exponents:
`2^(2x+1) = 2^(2x) *2 = 2*(2^x)^2 `
Then use substitution `y = 2^x` . Note that the y can have only positive values because any exponent of 2 is positive. The equation becomes
`3*y +2 = 2y^2`
Write this equation in standard form:
`2y^2 - 3y-2 = 0`
The left-hand side now can be factored by grouping:
`2y^2-3y-2 = 2y^2 - 4y + y - 2 = 2y(y-2) +(y-2) = (y-2)(2y+1)`
`(y-2)(2y+1) = 0`
y - 2 = 0 or 2y+1 = 0
y = 2 and y = -1/2
Since `y=2^x` it has to be positive, so the negative solution is extraneous.
Substitute the value of y = 2 to find x:
`2 = 2^x`
x = 1
Posted by ishpiro on June 11, 2013 at 1:48 AM (Answer #1)
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