Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals. x2e2x+2xe2x=8e2x

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Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.  x2e2x+2xe2x=8e2x

Nathanael Webber · Accepted Answer

Step 1    For solving Exponential Equations:    1) Isolate the exponential expression on one side of the equation.    2) Take the logarithm of each side, then use the Laws of Logarithms to &quot;bring down exponent&quot;.    3) Solve for the variable.    Step 2  Given,  x2e2x+2xe2x=8e2x  Add (−8e2x) on both sides,  x2e2x+2xe2x−8e2x=8e2x−8e2x  x2e2x+2xe2x−8e2x=0  Factor x2e2x+2xe2x−8e2x as e2x(x2+2x−8).  e2x(x2+2x−8)=0  Divide by e2x on both sides,  e2x(x2+2x−8)e2x=0e2x  x2+2x−8=0  Factor x2+2x−8 as (x+4)(x−2)  (x+4)(x−2)=0  By the zero factor property,  x+4=0 and x−2=0  Taken,  x+4=0  Add −4 on both sides,  x+4−4=0−4  x=−4  Now taken,  x−2=0  Add 2 on both sides,  x−2+2=0+2  x=2  Thus, the exact solution is x={−4, 2}.  Final statement:  Therefore, the exact solution is x={−4, 2}

Find the exact solution or use a calculator to approximate to two decimals x^{2}e^{2x}+2xe^{2x}=8e^{2x}

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