How to Find All Real Solutions of an Equation

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Frequently, in Algebra class, you will be called to find all "real solutions" of an equation. Such questions essentially are asking you to find all solutions of an equation, and should any imaginary solutions (containing the imaginary number 'i') come up, to discard these solutions. Therefore, most of the time, you will approach both equations with only real solutions and equations with both real and imaginary solutions the same way: find the solutions, and discard the ones that are not real numbers.

  • Simplify the equation as much as possible. For instance, if given the equation x4 + x2 - 6 = 0, you can use a u-substitution to simplify and then factor. If x2=u, then the equation becomes u2+u-6=0.

  • Factor the simplified equation. You can rewrite the equation in Step 1 as u2+3u-2u-6=0, then rewrite as u(u+3)-2(u+3)=0, which becomes (u-2)(u+3)=0.

  • Find the roots of the factored equation. Here, they are u=2 and u=3. Since x2=u, x must equal +/- sqrt(2), and +/- sqrt(3).

  • Discard any imaginary solutions, such as the square root of a negative number. Here, there are no imaginary solutions.

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