Answer :
Final answer:
To solve the equation x^4 - 4x^2 - 21 = 0, we can use the quadratic formula. By substituting x^2 with a new variable, we can simplify the equation and solve for the variable. The solutions to the equation are x = √7 and x = -√7.
Explanation:
To solve the equation x^4 - 4x^2 - 21 = 0, we can use a substitution. Let's substitute x^2 with a new variable, let's say y. The equation becomes y^2 - 4y - 21 = 0. Now we have a new quadratic equation that we can solve using factoring, completing the square, or the quadratic formula.
Let's solve it using the quadratic formula: y = (-(-4) ± √((-4)^2 - 4(1)(-21))) / (2(1)). Simplifying the equation gives us y = (4 ± √(16 + 84)) / 2, which further simplifies to y = (4 ± √100) / 2. Taking the square root of 100 gives us y = (4 ± 10) / 2, which gives us two possible solutions y = 7 and y = -3.
Since y was our substitution for x^2, we substitute back: x^2 = 7 and x^2 = -3. However, x^2 can't be a negative number, so we discard x^2 = -3. Solving x^2 = 7, we take the square root of both sides and get x = ±√7. Therefore, the solutions to the equation x^4 - 4x^2 - 21 = 0 are x = √7 and x = -√7.
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