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$$ 3^x = 4x^2 $$

My prof's notes just says to look at the graph and you'll see that they intersect in 3 points. But is there another way of doing this if you can't graph it?

Martin Sleziak
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TheEyesHaveIt
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  • You could define a function $f(x)=3^x-4x^2$ and look for zeros if you've worked any with finding zeros. – BeaumontTaz Jul 10 '14 at 14:19
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    You can say that there are at most 3 solutions, see http://math.stackexchange.com/questions/646092/number-of-solutions-of-px-eax-if-p-is-a-polynomial – gammatester Jul 10 '14 at 14:19
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    By inspection, you could easily locate three roots since, if $f(x)=3^x-4 x^2$, you have $f(-1)=-11/3$,$f(0)=1$,$f(1)=-1$,$f(2)=-7$,$f(3)=-9$,$f(4)=17$. – Claude Leibovici Jul 10 '14 at 14:20
  • @Claude; for those of us who are lazy, the $x$ values to look at are "large negative", $0$, $1$ and "large positive", giving $f(x)$ as "larger negative", $1-0=1$, $3-4=-1$, and "even larger positive" – Henry Jul 10 '14 at 14:28

2 Answers2

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The Intermediate Value Theorem migh be usefull here. Let $f(x) = 3^x - 4x^2$. Then $f(-1) <0$, $f(0) >0$, $f(1)<0$ and $f(4)>0$

As $f$ is continuous, there is at least one root in ]-1,0[, one root in ]0,1[ and one root in ]1,4[

Jonas Gomes
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Mathematica

Solve[3^x == 4 x^2, x]

There are no analytic solutions.

Martin Wang
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  • There are analytic solutions using Lambert function as for any equation which can write $A+Bx+C\log(Dx+E)=0$. in principle, any CAS provides the three solutions. – Claude Leibovici Jul 10 '14 at 14:53