What is the quadratic formula of $e^{2 x} - 2 e^{x} = 1$ $e^(2x) - 2e^x = 1$ ?

Question

What is the quadratic formula of $e^{2 x} - 2 e^{x} = 1$ $e^(2x) - 2e^x = 1$ ?

1 Answer

Impact of this question

4014 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-10-31T21:33:40

Recognise this as quadratic in $e^{x}$ $e^x$ and hence solve using the quadratic formula to find:

$x = ln (1 + \sqrt{2})$ $x = ln(1+sqrt(2))$

Explanation:

This is an equation that is quadratic in $e^{x}$ $e^x$ , rewritable as:

${(e^{x})}^{2} - 2 (e^{x}) - 1 = 0$ $(e^x)^2-2(e^x)-1 = 0$

If we substitute $t = e^{x}$ $t = e^x$ , we get:

$t^{2} - 2 t - 1 = 0$ $t^2-2t-1 = 0$

which is in the form $a t^{2} + b t + c = 0$ $at^2+bt+c = 0$ , with $a = 1$ $a=1$ , $b = - 2$ $b=-2$ and $c = - 1$ $c=-1$ .

This has roots given by the quadratic formula:

$t = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{2 \pm \sqrt{4 + 4}}{2} = 1 \pm \sqrt{2}$ $t = (-b+-sqrt(b^2-4ac))/(2a) = (2+-sqrt(4+4))/2 = 1+-sqrt(2)$

Now $1 - \sqrt{2} < 0$ $1-sqrt(2) < 0$ is not a possible value of $e^{x}$ $e^x$ for Real values of $x$ $x$ .

So $e^{x} = 1 + \sqrt{2}$ $e^x = 1 + sqrt(2)$ and $x = ln (1 + \sqrt{2})$ $x = ln(1+sqrt(2))$

Science

Math

Humanities

... and beyond

What is the quadratic formula of $e^{2 x} - 2 e^{x} = 1$ $e^(2x) - 2e^x = 1$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

What is the quadratic formula of e2x−2ex=1e^(2x) - 2e^x = 1?

1 Answer

Explanation:

Related questions

Impact of this question

What is the quadratic formula of $e^{2 x} - 2 e^{x} = 1$ $e^(2x) - 2e^x = 1$ ?