hi,
i nned to solve this equation:
x^3-5x=8
x=2.7
im unsure how to work this out.
i know i would do 2.7^3-13.5x=8 but i don't know what to do after this.
some help would be great!
thanks!
Let $\displaystyle f(x)=x^3-5x-8$ so that you have to find a value of $\displaystyle x$ that satisfies the equation $\displaystyle f(x)=0$. Observe that $\displaystyle f(2)<0$ and $\displaystyle f(3)>0$. So there lies x , (where $\displaystyle 2<x<3$,) such that f(x)=0.
See that $\displaystyle f(2.5) < 0$. So for $\displaystyle f(x)$ to be $\displaystyle 0$ , you have $\displaystyle 2.5<x<3$.
After this take $\displaystyle x=2.6,2.65,...$ so on and notice when $\displaystyle f(x)$ changes its sign. After afew couple of steps you will get the solution according to what precision you want to maintain.
However, the correct solution is $\displaystyle 2.8025$
To begin with, think in simple way. $\displaystyle x=2$ and $\displaystyle x-2=0$ are equivalent. Aren't they? So here solving $\displaystyle x^3-5x=8$ is equivalent to solving $\displaystyle x^3-5x-8=0$. Alright? You are just subtracting $\displaystyle 8$ from both the sides; so they must be equivalent.
Next try to understand what does a solution of an equation mean. I suppose you have the preliminary concept regarding function and graph. If you are gievn a function $\displaystyle f(x)$ (for example, here you have $\displaystyle f(x)=x^3-5x=8$), plot $\displaystyle f(x)$ in a graph ( that is, put values of $\displaystyle x$ along horizontal X-axis and values of $\displaystyle f(x)$ along vertical Y-axis). The function will have a solution at $\displaystyle x_0$ if $\displaystyle f(x)$is $\displaystyle 0$ at $\displaystyle x=x_0$ (that is $\displaystyle f(x_0)=0$). In this case, your function looks like THIS. The blue line denotes $\displaystyle f(x)$. Notice that as $\displaystyle x$ increases, $\displaystyle f(x)$ increases ( $\displaystyle f(x)$ becomes +ve from -ve). During this increase, the curve cuts the X-axis at some point (red point)-- that particular point is the solution of your equation.
So this is the concept. This is why you should find two consecutive values of $\displaystyle x_1 , x_2$ such that $\displaystyle f(x_1)<0$ and $\displaystyle f(x_2)>0$, because your required solution always lies in between $\displaystyle x_1$ and$\displaystyle x_2$
Now I hope you'll understand what my previous post meant.
The red point is the solution: 2.80259