To determine the behavior of the graph of the equation [tex]\( x^2 + 1 = 0 \)[/tex] in relation to the x-axis, we need to examine the roots of the quadratic equation.
The given quadratic equation is:
[tex]\[ x^2 + 1 = 0 \][/tex]
Let's rewrite it in standard form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 1 \)[/tex].
To determine the nature of the roots, we use the discriminant formula:
[tex]\[ D = b^2 - 4ac \][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[ D = 0^2 - 4(1)(1) = 0 - 4 = -4 \][/tex]
The discriminant [tex]\( D = -4 \)[/tex] is less than zero. This implies that the quadratic equation [tex]\( x^2 + 1 = 0 \)[/tex] has no real roots, only complex roots.
Since the discriminant is negative and there are no real roots, the graph of the equation [tex]\( x^2 + 1 = 0 \)[/tex] does not intersect or touch the x-axis.
Hence, the correct option is:
c) Neither touches nor intersects the x-axis.
