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PDF Editor FAQ

What is the value of [math]x[/math][math][/math] if [math]4^x+6^x=9^x[/math]?

You can actually solve this algebraically.We can rewrite the given equation as[math](2^x)^2+2^x\,3^x=(3^x)^2.[/math]Dividing both sides by [math](3^x)^2[/math], we obtain[math]\left(\left(\frac{2}{3}\right)^x\right)^2+\left(\frac{2}{3}\right)^x=1.[/math]Let [math]y=\left(\frac{2}{3}\right)^x[/math]. We have[math]y^2+y-1=0.[/math]This is a simple quadratic equation, with roots [math]\dfrac{-1\pm\sqrt{5}}{2}[/math]. Thus[math]\left(\dfrac{2}{3}\right)^x=\dfrac{-1+\sqrt{5}}{2}[/math] or [math]\left(\dfrac{2}{3}\right)^x=\dfrac{-1-\sqrt{5}}{2}.[/math]Let’s start with the positive root. We obtain[math]x=\dfrac{\ln\frac{-1+\sqrt{5}}{2}}{\ln\frac{2}{3}}=\dfrac{\ln(\sqrt{5}-1)-\ln 2}{\ln 2-\ln 3}=\mathbf{1.18681439}.[/math]The negative root yields[math]x=\dfrac{\ln\frac{-1-\sqrt{5}}{2}}{\ln\frac{2}{3}}=\dfrac{\ln\frac{1+\sqrt{5}}{2}+i\,\pi}{\ln\frac{2}{3}}=\mathbf{-1.18681439-7.74812084i}.[/math](Technically, we have infinitely many solutions, but I just gave the principal solution for each equation.)

If [math]p(x)=\int \frac{dx}{x+x^7}[/math], then what is [math] \int \frac{x^6}{x+x^7} dx[/math] equal to?

Let[math]\displaystyle p(x) = \int \frac{1}{x+x^7} \, dx \text{ and } q(x) = \int \frac{x^6}{x+x^7} \, dx. \tag*{}[/math]Adding these together yields[math]\begin{align*} p(x) + q(x) &= \displaystyle \int \frac{1 + x^6}{x+x^7} \, dx\\ &= \displaystyle \int \frac{1 + x^6}{x(1 +x^6)} \, dx\\ &= \displaystyle \int \frac{1}{x} \, dx\\ &= \ln|x| + C. \end{align*} \tag*{}[/math]Therefore, we conclude that[math]q(x) = \displaystyle \int \frac{x^6}{x+x^7} \, dx = -p(x) + \ln|x| + C. \tag*{}[/math]

When 18 is added to a number and then divided by 4, the answer is 6. What is that number?

let that number be x,so when 18 is added to x and then divided by 4, the answer is 6 can also be written as;(18+x) /4 =618 + x = 4 x 618 + x = 24x = 24 - 18x = 6The number is 6 (six)

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