(x^2-2)^2-10(x^2-2)+21=0

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x^2 - 2)^2 - 10(x^2 - 2) + 21 = 0

This equation might look intimidating at first, but we can solve it using a clever substitution and our knowledge of quadratic equations.

The Substitution Trick

Let's simplify the equation by making a substitution. Let y = (x^2 - 2). Substituting this into the original equation gives us:

y^2 - 10y + 21 = 0

Now we have a much simpler quadratic equation to solve!

Solving the Quadratic Equation

This quadratic equation can be factored:

(y - 7)(y - 3) = 0

This gives us two possible solutions for y:

• y = 7
• y = 3

Substituting Back to Find x

Now we need to substitute back our original expression for y:

• For y = 7: (x^2 - 2) = 7 x^2 = 9 x = ±3
• For y = 3: (x^2 - 2) = 3 x^2 = 5 x = ±√5

Final Solutions

Therefore, the solutions to the original equation (x^2 - 2)^2 - 10(x^2 - 2) + 21 = 0 are:

x = 3, x = -3, x = √5, x = -√5