Asked by htht
What are the solutions of the equation 9x4 – 2x2 – 7 = 0? Use u substitution to solve.
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GPT-5 mini
AI
Let u = x^2. Then the equation becomes
9u^2 − 2u − 7 = 0.
Solve the quadratic:
Discriminant Δ = (−2)^2 − 4·9·(−7) = 4 + 252 = 256, sqrt Δ = 16.
u = [2 ± 16]/(18) = {1, −7/9}.
Now return to x:
- If x^2 = 1, x = ±1.
- If x^2 = −7/9, x = ± i·√7/3.
So the four solutions are x = 1, x = −1, x = (i√7)/3, x = −(i√7)/3. The real solutions are x = ±1.
9u^2 − 2u − 7 = 0.
Solve the quadratic:
Discriminant Δ = (−2)^2 − 4·9·(−7) = 4 + 252 = 256, sqrt Δ = 16.
u = [2 ± 16]/(18) = {1, −7/9}.
Now return to x:
- If x^2 = 1, x = ±1.
- If x^2 = −7/9, x = ± i·√7/3.
So the four solutions are x = 1, x = −1, x = (i√7)/3, x = −(i√7)/3. The real solutions are x = ±1.
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