How do you explain which if x is within the quadratic graph or outside in inequalities?

Question

In inequalities lets say for example x^2-3x+10 is greater than 0. So a sketched quadratic graph of this would show that it crosses through -2 and 5 on the x axis. How do I know if x is less than or greater than -2 similarly if it is less than or greater than 5 on this what is an easy way of calculating this out? Also what will be the difference if the quadratic is less than 0?

Answer 1

In the graph of $y=ax^2+bx+c$ for example, note that for every $x$ the corresponding value of $y$ on the graph displays the value of the quadratic expression at that point. So if the graph of the quadratic shows that for a certain $x$ the graph is above the $x$-axis, then that means that $y>0$ or $ax^2+bx+c>0$.

Example. The solution to the inequality $x^2-17x+70\geq0$ is $x\leq7$ or $x\geq10$. This is because $x^2-17x+70=(x-7)(x-10)$, so the graph $y=x^2-17x+70$ crosses the $x$-axis at $x=7$ and $x=10$. The leading coefficient is positive, so the graph looks like the following:

Observe that when $x\leq7$, the graph is above the $x$-axis, so the quadratic is $\geq0$. Similarly this is the case when $x\geq10$. On the other hand when $7<x<10$, the graph is below the $x$-axis, so $x^2-17x+70<0$.