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Wittaler [7]
3 years ago
15

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

Mathematics
1 answer:
Setler [38]3 years ago
3 0

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:

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