Answer
(a) The graph is as shown.
(b) The $x$-intercepts are $(1.5,0)$.
(c) The solution is: $x=1.5$
(d) The result in part (c) is the same with the $x$-intercepts of the graph.
Work Step by Step
(a) The graph is as shown.
(b) The $x$-intercepts are $(1.5,0)$.
(c) Setting $y=0$:
$$0=2x-\sqrt{15-4x}$$
$$2x=\sqrt{15-4x}$$
Squaring both sides:
$$4x^2=15-4x$$
$$4x^2+4x-15=0$$
Using quadratic formula:
$$x=\frac{-4\pm\sqrt{4^2-4(4)(-15)}}{2(4)}=-\frac{1}{2}\pm2$$
$$x_1=-\frac{1}{2}+2=\frac{3}{2}=1.5$$
Checking:
$$0=2(1.5)-\sqrt{15-4(1.5)}$$
$$0=0~True$$
Thus, $x=1.5$ is a solution.
$$x_2=-\frac{1}{2}-2=-\frac{5}{2}=-2.5$$
Checking:
$$0=2(-2.5)-\sqrt{15-4(-2.5)}$$
$$0=-5-\sqrt5~False$$
Thus, $x=-2.5$ is not a solution.
Therefore, the solution is:
$$x=1.5$$
(d) The result in part (c) is the same with the $x$-intercepts of the graph.
