x^4 + 19x^2 - 20 = 0

Steps to Solve

1. Make a Substitution
   Let's make a substitution to simplify the equation. Let ( y = x^2 ). Then the original equation becomes:
   $$ y^2 + 19y - 20 = 0 $$

2. Use the Quadratic Formula
   We will solve for ( y ) using the quadratic formula, which is given by:
   $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
   For our equation, ( a = 1 ), ( b = 19 ), and ( c = -20 ). Plugging these values into the formula gives us:
   $$ y = \frac{-19 \pm \sqrt{19^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} $$

3. Calculate the Discriminant
   Now we will compute the discriminant ( 19^2 - 4 \cdot 1 \cdot (-20) ):
   $$ 19^2 = 361 $$
   $$ -4 \cdot 1 \cdot (-20) = 80 $$
   Thus, the discriminant is:
   $$ 361 + 80 = 441 $$

4. Solve for y
   Now substitute the discriminant back into the quadratic formula:
   $$ y = \frac{-19 \pm \sqrt{441}}{2} $$
   Since ( \sqrt{441} = 21 ), we get:
   $$ y = \frac{-19 \pm 21}{2} $$

5. Find the Possible Values for y
   Calculating the two values will give us:
   $$ y_1 = \frac{2}{2} = 1 $$
   $$ y_2 = \frac{-40}{2} = -20 $$

6. Re-substitute to Find x
   Now we convert back to ( x ) using the substitution ( y = x^2 ):
   From ( y_1 = 1 ):
   $$ x^2 = 1 \implies x = \pm 1 $$
   From ( y_2 = -20 ):
   Since ( x^2 ) cannot be negative, there are no real solutions from this case.

7. List the Final Solutions
   Thus, the final solutions to the equation ( x^4 + 19x^2 - 20 = 0 ) are:
   $$ x = 1 \text{ and } x = -1 $$