Answer:
1. [tex]x_1=2\sqrt{2}\\x_2=-2\sqrt{2}\\x_3=i2\sqrt{2}\\x_4=-i2\sqrt{2}[/tex]
2. [tex]A=(2\sqrt{2},0)[/tex] and [tex]B=(-2\sqrt{2},0)[/tex]
Step-by-step explanation:
We have the expression [tex]x^4-64=0[/tex] to find the solutions we have to clear [tex]x[/tex]. Then,
[tex]x^4-64=0[/tex] Add 64 in both sides of the equation.
[tex]x^4-64+64=64\\x^4=64[/tex]
Now we have to re-write the equation:
[tex]u=x^2[/tex]⇒[tex]u^2=(x^2)^2\\u^2=x^4[/tex]
Then,
[tex]u^2=64[/tex]
Apply square root on both members
[tex]\sqrt{u^2}=\sqrt{64}\\u=8 , u=-8[/tex]
Now substitute back [tex]u=x^2[/tex]
1. [tex]x^2=8[/tex]
2. [tex]x^2=-8[/tex]
Solving for x:
1. [tex]x=+\sqrt{8}=\sqrt{2^3}=\sqrt{2^2}.\sqrt{2}\\x=2\sqrt{2}\\or\\x=-\sqrt{8}\\x=-2\sqrt{2}[/tex]
2. [tex]x=+\sqrt{-8}=\sqrt{(-1).8} =\sqrt{-1}\sqrt{8} \\x=i\sqrt{8}\\x=i.2\sqrt{2} \\or\\x=-\sqrt{-8}\\x=-i.2\sqrt{2}[/tex]
Then the solutions for [tex]x^4-64=0[/tex] are:
[tex]x_1=2\sqrt{2}\\x_2=-2\sqrt{2}\\x_3=i2\sqrt{2}\\x_4=-i2\sqrt{2}[/tex]
To find the x intercepts of the graph [tex]y=x^4-64[/tex] we have to replace with y=0.
This means: [tex]x^4-64=0[/tex]
We already found the solutions for the expression, but we have to consider only the real solutions.
[tex]x_1=2\sqrt{2}\\x_2=-2\sqrt{2}[/tex]
Because in a graph we can't have imaginary solutions.
Then the x intercepts of the graph are:
[tex]A=(2\sqrt{2},0)[/tex] and [tex]B=(-2\sqrt{2},0)[/tex]
The graph of the function is:
