Someone at work sent me this algebra puzzle, which I think came from The Times newspaper:

Solve 9ˣ+15ˣ=25ˣ.

(Put a CW on solutions in replies, please!)

Have you seen it before? Did you see it in The Times?

· · Web · · ·

solution

Here's my working:

Want to solve for $$x$$:

$9^x + 15^x = 25^x$

First guess: it's a little bit less than $$1$$:

$\begin{eqnarray*} 9^0 + 15^0 &=& 2 &>& 1 = 25^0 \\ 9^1 + 15^1 &=& 24 &<& 25 = 25^1 \end{eqnarray*}$

Do some rearranging:

$9^x+15^x=25^x$
$1+ \left(\frac{5}{3}\right)^x = \left(\frac{25}{9}\right)^x$ (divide by $$9^x$$)
$\left(\frac{25}{9}\right)^x - \left(\frac{5}{3}\right)^x - 1 = 0$
$\left(\frac{5}{3}\right)^{2x} - \left(\frac{5}{3}\right)^x - 1 = 0$
...

solution

...
$\left(\frac{5}{3}\right)^x = \frac{1 \pm \sqrt{5}}{2}$ (quadratic formula)
$\left(\frac{5}{3}\right)^x = \frac{1 + \sqrt{5}}{2}$ (only one root is real)
$x \ln \left(\frac{5}{3}\right) = \ln\left(\frac{1 + \sqrt{5}}{2}\right)$ (take logs)
$x \left(\ln5 - \ln3\right) = \ln\left(1 + \sqrt{5}\right) - \ln2$
$x = \frac{\ln\left(1 + \sqrt{5}\right) - \ln2}{\ln5 - \ln3}$

solution

So my question is: was this set up so that the golden ratio ϕ=(1+√5)/2 would turn up, or could you cook up a similar puzzle without it?

solution

@christianp it seems to me 3 and 5 could be any numbers a and b, but you have to use a*a, a*b, and b*b. For any a and b you would still have phi show up.

@christianp I've seen something very like this, and I suspect someone took the basic idea (seen somewhere else) I created the specific example to have the Easter Egg.

Nice one.

@christianp
I may be stupid but i only see 0 as a solution... 24x =25x

@oldsysops The ˣ means "to the power of x", not "multiplied by x".
So with x=0, you get 9⁰+15⁰ = 1+1 = 2, and 25⁰ =1, so the equation isn't satisfied.

@christianp
So yes, i do have misunderstood the equation.
thanks. 🙂 The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!