Question

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

Answer 1

1.
`x_1=2√(2)\\x_2=-2√(2)\\x_3=i2√(2)\\x_4=-i2√(2)`

2.
`A=(2√(2),0)` and
`B=(-2√(2),0)`

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

`√(u^2)=√(64)\\u=8 , u=-8`

Now substitute back
`u=x^2`

1.
`x^2=8`

2.
`x^2=-8`

Solving for x:

1.
`x=+√(8)=√(2^3)=√(2^2).√(2)\\x=2√(2)\\or\\x=-√(8)\\x=-2√(2)`

2.
`x=+√(-8)=√((-1).8) =√(-1)√(8) \\x=i√(8)\\x=i.2√(2) \\or\\x=-√(-8)\\x=-i.2√(2)`

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

`x_1=2√(2)\\x_2=-2√(2)\\x_3=i2√(2)\\x_4=-i2√(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√(2)\\x_2=-2√(2)`

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

Then the x intercepts of the graph are:

`A=(2√(2),0)` and
`B=(-2√(2),0)`

The graph of the function is: