Question

Find the solutions to x⁴-64=0 and the x-intercepts of the graph of y=x⁴-64.

Answer 1

Answer:

1. $x_1=2\sqrt{2}\\x_2=-2\sqrt{2}\\x_3=i2\sqrt{2}\\x_4=-i2\sqrt{2}$

2. $A=(2\sqrt{2},0)$ and $B=(-2\sqrt{2},0)$

Step-by-step explanation:

We have the expression $x^4-64=0$ to find the solutions we have to clear $x$. Then,

$x^4-64=0$ Add 64 in both sides of the equation.

$x^4-64+64=64\\x^4=64$

Now we have to re-write the equation:

$u=x^2$⇒$u^2=(x^2)^2\\u^2=x^4$

Then,

$u^2=64$

Apply square root on both members

$\sqrt{u^2}=\sqrt{64}\\u=8 , u=-8$

Now substitute back $u=x^2$

1. $x^2=8$

2. $x^2=-8$

Solving for x:

1. $x=+\sqrt{8}=\sqrt{2^3}=\sqrt{2^2}.\sqrt{2}\\x=2\sqrt{2}\\or\\x=-\sqrt{8}\\x=-2\sqrt{2}$

2. $x=+\sqrt{-8}=\sqrt{(-1).8} =\sqrt{-1}\sqrt{8} \\x=i\sqrt{8}\\x=i.2\sqrt{2} \\or\\x=-\sqrt{-8}\\x=-i.2\sqrt{2}$

Then the solutions for $x^4-64=0$ are:

$x_1=2\sqrt{2}\\x_2=-2\sqrt{2}\\x_3=i2\sqrt{2}\\x_4=-i2\sqrt{2}$

To find the x intercepts of the graph $y=x^4-64$ we have to replace with y=0.

This means: $x^4-64=0$

We already found the solutions for the expression, but we have to consider only the real solutions.

$x_1=2\sqrt{2}\\x_2=-2\sqrt{2}$

Because in a graph we can't have imaginary solutions.

Then the x intercepts of the graph are:

$A=(2\sqrt{2},0)$ and $B=(-2\sqrt{2},0)$

The graph of the function is: