(x^2-7)^2-6(x^2-7)-16=0

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x^2 - 7)^2 - 6(x^2 - 7) - 16 = 0

This equation might look intimidating at first, but we can solve it using a simple substitution method.

1. Substitution

Let's substitute y = x^2 - 7. This will simplify our equation:

y^2 - 6y - 16 = 0

2. Factoring the Quadratic

Now we have a standard quadratic equation. We can factor it:

(y - 8)(y + 2) = 0

This gives us two possible solutions for y:

• y = 8
• y = -2

3. Substitute Back

Now, let's substitute back x^2 - 7 for y in each solution:

• For y = 8: x^2 - 7 = 8 x^2 = 15 x = ±√15

• For y = -2: x^2 - 7 = -2 x^2 = 5 x = ±√5

4. Solutions

Therefore, the solutions to the equation (x^2 - 7)^2 - 6(x^2 - 7) - 16 = 0 are:

x = √15, x = -√15, x = √5, x = -√5