Answer: The parabola does not intercept the x-axis.
Step-by-step explanation:
The parabola intercepts the x-axis when [tex]y=0[/tex], then, you need to substitute [tex]y=0[/tex] into the equation:
[tex]y=4x^2 -4x+9\\0=4x^2 -4x+9[/tex]
Now, use the Quadratic formula:
[tex]x=\frac{-b\±\sqrt{b^2-4ac}}{2a}[/tex]
In this case:
[tex]a=4\\b=-4\\c=9[/tex]
Substituting these values and evaluating, you get:
[tex]x=\frac{-(-4)\±\sqrt{(-4)^2-4(4)(9)}}{2(4)}\\\\x=\frac{4\±\sqrt{-128}}{8}[/tex]
Remeber that:
[tex]\sqrt{-1}=i[/tex]
Then, rewriting:
[tex]x=\frac{4\±8i\sqrt{2}}{8}[/tex]
Simplifying:
[tex]x=\frac{4(1\±2i\sqrt{2}}{4(2)}\\\\x=\frac{1\±2i\sqrt{2}}{2}\\\\x=\frac{1}{2}\±\frac{2i\sqrt{2}}{2}\\\\x=\frac{1}{2}\±i\sqrt{2}[/tex]
Then:
[tex]x_1=\frac{1}{2}+i\sqrt{2}[/tex]
[tex]x_2=\frac{1}{2}-i\sqrt{2}[/tex]
The roots are complex, therefore, the parabola does not intercept the x-axis.
