Question

1/2𝑥^5 =1/5⁢𝑥^2 − 2 solve by graphing

Answer 1

You want the intersections of
y1 = (1/2)x^5 and y2 = (1/5)x^2 − 2.
Equivalently solve
(1/2)x^5 − (1/5)x^2 + 2 = 0.
Multiply by 10 to clear fractions:
5x^5 − 2x^2 + 20 = 0.

Define f(x) = (1/2)x^5 − (1/5)x^2 + 2. By graphing or sign changes you can bracket the root:
- f(0) = 2 (>0)
- f(−1.3) ≈ −0.194 ( <0)
- f(−1.25) ≈ 0.162 (>0)

So there is a root between −1.3 and −1.25. Refining (or using a solver) gives the unique real solution
x ≈ −1.274 (to three decimal places).

(The derivative f'(x) = (5/2)x^4 − (2/5)x is positive for every x < 0, so f is strictly increasing on (−∞,0) and there is exactly one negative root; f stays positive for x ≥ 0, so no positive roots.)

The intersection point (x, y) is approximately (−1.274, −1.676) (y ≈ (1/5)x^2 − 2).

If you want, I can produce a plotted graph or refine the root to more digits.