Q. 18 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.    

$x^2+2x+4>0$

Step-by-Step Solution

Answer

The given inequality has no solution. 

Step 1. Given information

The given inequality is $x^2+2x+4>0$

Step 2. Find the intercepts and vertex

• The value of $f\left(0\right)=4$.
• So, the y-intercept is $(0,4)$.
• There are no real roots of $x^2+2x+4=0$, so, there are no x-intercepts of the curve.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-2}{2\left(1\right)}=-1$.
• Find the value of the function at $x=-1$.

$\begin{array}{rcl}f(-1)& =& {(-1)}^2+2(-1)+4\\ & =& 1-2+4\\ & =& 3\end{array}$

• So, the vertex is $(-1,3)$.

Step 3. Plot the graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.     

Step 4. Find the solution set

• From the graph, there is no region that lies below the x-axis.
• So, the given inequality has no solution.