Question

Show that the equation x cubed + 8x = 30 has a solution between x = 2.2 and x =2.3 ?

Answer 1

$x^3+8x=30\\ x^3+8x-30=0\\\\ 2.2^3+8\cdot2.2-30=10.648+17.6-30=-1.752\\ 2.3^3+8\cdot2.3-30=12.167+18.4-30=0.567$

For $x=2.2$ the value of $x^3+8x-30$ is negative, and for $x=2.3$ the value is positive which means between $x=2.2$ and $x=2.3$ there is an x for which the value of $x^3+8x-30$ is equal to 0 and therefore there is a solution.