solve the exponential equation express in exact form only show all work.
2e^x= squareroot 10e^x +3
sqrt of what ???
sorry about that squre root (10e^x) +3
To solve the exponential equation 2e^x = √(10e^x) + 3, we will follow these steps:
Step 1: Start by rearranging the equation to one side.
2e^x - √(10e^x) - 3 = 0
Step 2: Let's make a substitution to simplify the equation.
Let's substitute y = e^x.
Substituting y, we have:
2y - √(10y) - 3 = 0
Step 3: Now, we can rewrite the equation in terms of y.
√(10y) = 2y - 3
Step 4: Square both sides of the equation to eliminate the square root.
(√(10y))^2 = (2y - 3)^2
10y = (2y - 3)^2
Step 5: Expand the right side of the equation.
10y = 4y^2 - 12y + 9
Step 6: Move all terms to one side to form a quadratic equation.
4y^2 - 22y + 9 = 0
Step 7: To solve this quadratic equation, we can use the quadratic formula:
y = [-b ± √(b^2 - 4ac)] / 2a
For our quadratic equation 4y^2 - 22y + 9 = 0, the values of a, b, and c are:
a = 4, b = -22, and c = 9.
Substituting the values in the quadratic formula:
y = [-( -22) ± √((-22)^2 - 4(4)(9))] / (2 * 4)
Simplifying:
y = [22 ± √(484 - 144)] / 8
y = [22 ± √340] / 8
y = [22 ± 2√85] / 8
Step 8: Simplify the expression further.
y = (11 ± √85) / 4
Step 9: Recall that we initially substituted y = e^x.
So, we have:
e^x = (11 ± √85) / 4
Step 10: Solve for x by taking the natural logarithm (ln) of both sides.
ln(e^x) = ln((11 ± √85) / 4)
x = ln((11 ± √85) / 4)
Hence, x = ln((11 ± √85) / 4) is the exact form solution of the given exponential equation.