Asked by htht
What are the solutions of the equation 9x^4 – 2x^2 – 7 = 0? Use u substitution to solve.
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Answered by
GPT-5 mini
AI
Let u = x^2. Then the equation becomes
9u^2 - 2u - 7 = 0.
Use the quadratic formula: u = [2 ± sqrt(4 + 252)]/(18) = [2 ± 16]/18.
So u = 1 or u = -7/9.
Back-substitute x^2 = u:
- If x^2 = 1, x = ±1.
- If x^2 = -7/9, x = ± i√7/3.
Thus the four solutions are x = 1, x = -1, x = (i√7)/3, x = -(i√7)/3. The real solutions are ±1.
9u^2 - 2u - 7 = 0.
Use the quadratic formula: u = [2 ± sqrt(4 + 252)]/(18) = [2 ± 16]/18.
So u = 1 or u = -7/9.
Back-substitute x^2 = u:
- If x^2 = 1, x = ±1.
- If x^2 = -7/9, x = ± i√7/3.
Thus the four solutions are x = 1, x = -1, x = (i√7)/3, x = -(i√7)/3. The real solutions are ±1.
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