Rearrange an equation that is quadratic in form into its standard form, which is a( )^2 + b( ) + c = 0. The variables "a" and "b" represent coefficients and "c" represents the constant. For example, use the equation x^4 - x^2 = 2. Subtract 2 from both sides of the equation to move 2 to the left side of the equation. This results in x^4 - x^2 - 2 = 2 - 2, which leaves x^4 - x^2 - 2 = 0.
Rewrite the variables and their exponents in the equation so that the equation matches the standard quadratic form in which the first term contains an exponent of 2 and the second term contains no exponent. For example, rewrite x^4 as (x^2)^2. This leaves (x^2)^2 - (x^2) - 2 = 0.
Substitute the variable "n" for the part of the second term in parentheses next to "b." For example, x^2 is the part of the second term in parentheses, so substitute n for each x^2 in the equation. This leaves n^2 - n - 2 = 0, which is a quadratic equation in which n = x^2.
Factor the quadratic equation by finding the two two-term expressions that equal the equation when multiplied together. Determine the first terms of each expression that, when multiplied together, equal the first term in the quadratic equation. For example, n times n equals n^2, the first term of the equation. Thus, n is the first term of each factor expression. Set up your factor expressions like this: (n )(n ).
Determine the second terms of each factor expression that equal the constant of the equation when multiplied together and equal the coefficient of the second term in the equation when added together. For example, 1 and -2 equal the constant, -2, when multiplied and equal the coefficient of the second term, -1, when added together. Therefore, 1 and -2 are the second terms of the factor expressions. The factored equation is (n + 1)(n - 2) = 0.
Set the first factor equal to 0 and solve for the variable. This results in n + 1 = 0. Subtract 1 from both sides to solve for n, which results in n = -1.
Set the second factor equal to 0 and solve for the variable. This results in n - 2 = 0. Add 2 to both sides to solve for n, which results in n = 2. Therefore, n equals -1 and 2.
Set the first result equal to the variable that n represents in the quadratic equation and solve for the variable. For example, n represents x^2 in the equation, so -1 = x^2. Find the positive and negative square root of -1 to solve for x: x equals the imaginary numbers i and -i. These allow you to take the square root of a negative number, which has no real number solution. You get a result of -1 if you square either i or -i.
Set the second result equal to the variable that n represents in the quadratic equation and solve for the variable. This results in 2 = x^2. Find the positive and negative square root of 2 to solve for x. This equals √2 and -√2. Therefore, x equals i, -i, √2 and -√2, which are the four solutions to the equation.
Substitute the first solution into the original equation and solve to verify that it generates a true equation. For example, (i)^4 - (i)^2 = 2, which leaves 1 - (-1) = 2. This solves to 2 = 2, which is a true equation.
Substitute the second solution into the original equation and solve to verify that it generates a true equation. For example, (-i)^4 - (-i)^2 = 2, which leaves 1 - (-1) = 2. The result is 2 = 2, which is a true equation.
Substitute the third solution into the original equation and solve. For example, this results in (√2)^4 - (√2)^2 = 2, which leaves 4 - 2 = 2. This leaves 2 = 2, which is a true equation.
Substitute the third solution into the original equation and solve. For example, this results in (-√2)^4 - (-√2)^2 = 2, which leaves 4 - 2 = 2. This solves to 2 = 2, which is a true equation. Therefore, all four solutions are correct.