Asked by bob
solve by graphing 𝑥^8 =−𝑥^7 +3
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GPT-5 mini
AI
Rearrange to find intersection points:
x^8 = −x^7 + 3 ⇔ x^8 + x^7 − 3 = 0.
Let f(x) = x^8 + x^7 − 3.
Observe f'(x) = x^6(8x+7). Since x^6 ≥ 0 for all x:
- f'(x) < 0 on (−∞, −7/8), so f is strictly decreasing there (at most one root).
- f'(x) ≥ 0 on [−7/8, ∞), so f is nondecreasing there (at most one root).
Because f(−∞) → +∞, f(0) = −3 < 0 and f(∞) → +∞, there are exactly two real roots: one with x < −7/8 and one with x > 0.
Numerically (by bisection/Newton or by graphing), the two real solutions are approximately:
x ≈ −1.357 and x ≈ 1.055
(You can refine these further; to four decimal places they are about x ≈ −1.3566 and x ≈ 1.0555.)
x^8 = −x^7 + 3 ⇔ x^8 + x^7 − 3 = 0.
Let f(x) = x^8 + x^7 − 3.
Observe f'(x) = x^6(8x+7). Since x^6 ≥ 0 for all x:
- f'(x) < 0 on (−∞, −7/8), so f is strictly decreasing there (at most one root).
- f'(x) ≥ 0 on [−7/8, ∞), so f is nondecreasing there (at most one root).
Because f(−∞) → +∞, f(0) = −3 < 0 and f(∞) → +∞, there are exactly two real roots: one with x < −7/8 and one with x > 0.
Numerically (by bisection/Newton or by graphing), the two real solutions are approximately:
x ≈ −1.357 and x ≈ 1.055
(You can refine these further; to four decimal places they are about x ≈ −1.3566 and x ≈ 1.0555.)
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