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Solve for x: x^{4} - 21x^{2} - 35 = 65

Example

Solution

Step 1 Write the equation in standard form.

Subtract 65 from both sides.

Step 2 Factor.

Step 3 Use the Zero Product Property.

Step 4 Solve for the variable.

x^{4} - 21x^{2} - 35

x^{4} - 21x^{2} - 100

(x^{2}
+ 4)(x^{2} - 25)

x^{2} + 4 = 0 or x^{2}
-25

x^{2} = - 4 or x^{2}

= 65

= 0

= 0

= 0

= 25

So, there are four solutions: -2i, +2i, -5, and +5.

The equation x^{4} - 21x^{2} - 35 = 65 written in standard form is x^{4} - 21x^{2} - 100
= 0. The graph of the corresponding function,
f(x) = x^{4} - 21x^{2} - 100 is shown.

The graph crosses the x-axis at only two locations, x = -5 and x
= 5. This is because these are the only real number solutions.