which graph represents the solution set for the quadratic inequality $x^{2}+2x + 1>0$?
which graph represents the solution set for the quadratic inequality $x^{2}+2x + 1>0$?
Answer
Explanation:
Step1: Factor the quadratic expression
We know that (x^{2}+2x + 1=(x + 1)^{2}) by the perfect - square formula ((a + b)^{2}=a^{2}+2ab + b^{2}), where (a=x) and (b = 1). So the inequality becomes ((x + 1)^{2}>0).
Step2: Solve the inequality
The equation ((x + 1)^{2}=0) has a root (x=-1). A non - zero real number squared is always positive. So the solution of the inequality ((x + 1)^{2}>0) is (x\neq - 1). On a number line, this is represented by an open circle at (x=-1) and shading to the left and right of (x=-1).
Answer:
The graph with an open circle at (x =-1) and shading to the left and right of (x=-1) (the third graph option in the order presented in the problem).