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Solve for x: x4 - 21x2 - 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. x4 - 21x2 - 35   x4 - 21x2 - 100(x2 + 4)(x2 - 25)x2 + 4 = 0 or x2 -25x2 = - 4 or x2 = 65  = 0= 0= 0= 25

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

The equation x4 - 21x2 - 35 = 65 written in standard form is x4 - 21x2 - 100 = 0. The graph of the corresponding function, f(x) = x4 - 21x2 - 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.