(x+7)^4-17(x+7)^2+16=0

Jasonbradley

2 min read

·

Jun 17, 2024

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Solving the Equation (x+7)^4 - 17(x+7)^2 + 16 = 0

This equation looks complicated at first glance, but we can simplify it using a clever substitution.

Substitution

Let's make the substitution y = (x+7). This will transform our equation into a quadratic:

y^4 - 17y^2 + 16 = 0

Solving the Quadratic

Now we have a standard quadratic equation in terms of 'y'. We can factor this equation as:

(y^2 - 1)(y^2 - 16) = 0

This gives us two possible solutions:

• y^2 - 1 = 0
  • y^2 = 1
  • y = ±1
• y^2 - 16 = 0
  • y^2 = 16
  • y = ±4

Back Substitution

Now that we have values for 'y', we can substitute back to find the solutions for 'x'. Remember, we defined y = (x+7):

• y = 1:
  • 1 = x + 7
  • x = -6
• y = -1:
  • -1 = x + 7
  • x = -8
• y = 4:
  • 4 = x + 7
  • x = -3
• y = -4:
  • -4 = x + 7
  • x = -11

Solution

Therefore, the solutions to the equation (x+7)^4 - 17(x+7)^2 + 16 = 0 are:

x = -6, -8, -3, -11