Question

which statement describes the graph of $f(x) = 4x^2 + 20x + 25?$○ the graph does not intersect the x-axis.○ the graph touches the x-axis at $(-2.5, 0)$.○ the graph intersects the x-axis at $(-0.4, 0)$ and $(0.4, 0)$.○ the graph intersects the x-axis at $(2, 0)$ and $(5, 0)$.

Explanation:

Step1: Find discriminant of quadratic

The discriminant of $ax^2+bx+c$ is $b^2-4ac$. For $f(x)=4x^2+20x+25$, $a=4$, $b=20$, $c=25$.
$\Delta = 20^2 - 4\times4\times25 = 400 - 400 = 0$

Step2: Solve for x-intercept

A discriminant of 0 means one repeated root. Use quadratic formula $x=\frac{-b}{2a}$.
$x = \frac{-20}{2\times4} = \frac{-20}{8} = -2.5$

Answer:

The graph touches the x-axis at (-2.5, 0).