1. **Problem:** Solve the inequality $$x^4 + 35x^2 - 36 \geq 0$$. 2. **Formula and rules:** To solve polynomial inequalities, first find the roots by setting the expression equal to zero, then test intervals between roots to determine where the inequality holds. 3. **Step 1: Set the expression equal to zero to find roots:** $$x^4 + 35x^2 - 36 = 0$$ 4. **Step 2: Use substitution:** Let $$y = x^2$$, then the equation becomes: $$y^2 + 35y - 36 = 0$$ 5. **Step 3: Solve the quadratic in $$y$$:** Using the quadratic formula: $$y = \frac{-35 \pm \sqrt{35^2 - 4 \times 1 \times (-36)}}{2} = \frac{-35 \pm \sqrt{1225 + 144}}{2} = \frac{-35 \pm \sqrt{1369}}{2}$$ 6. **Step 4: Calculate the discriminant:** $$\sqrt{1369} = 37$$ 7. **Step 5: Find the roots for $$y$$:** $$y_1 = \frac{-35 + 37}{2} = 1$$ $$y_2 = \frac{-35 - 37}{2} = -36$$ 8. **Step 6: Recall $$y = x^2$$, so $$x^2 = 1$$ or $$x^2 = -36$$ (discard since $$x^2$$ cannot be negative). 9. **Step 7: Find $$x$$ values:** $$x = \pm 1$$ 10. **Step 8: Test intervals around $$x = -1$$ and $$x = 1$$ to determine where $$x^4 + 35x^2 - 36 \geq 0$$: - For $$x < -1$$, pick $$x = -2$$: $$(-2)^4 + 35(-2)^2 - 36 = 16 + 140 - 36 = 120 > 0$$ - For $$-1 < x < 1$$, pick $$x = 0$$: $$0 + 0 - 36 = -36 < 0$$ - For $$x > 1$$, pick $$x = 2$$: $$16 + 140 - 36 = 120 > 0$$ 11. **Step 9: Write the solution:** $$x \in (-\infty, -1] \cup [1, \infty)$$ **Final answer:** $$x \leq -1$$ or $$x \geq 1$$.