1. **State the problem:** Solve for $x$ in the equation $$2(3x + 6 - x^2) = 7x + 5$$ to 3 significant figures. 2. **Expand and simplify:** Distribute the 2 on the left side: $$2 \times 3x + 2 \times 6 - 2 \times x^2 = 7x + 5$$ which gives $$6x + 12 - 2x^2 = 7x + 5$$ 3. **Bring all terms to one side:** $$6x + 12 - 2x^2 - 7x - 5 = 0$$ Simplify like terms: $$-2x^2 + (6x - 7x) + (12 - 5) = 0$$ $$-2x^2 - x + 7 = 0$$ 4. **Rewrite in standard quadratic form:** $$-2x^2 - x + 7 = 0$$ Multiply both sides by $-1$ to make the leading coefficient positive: $$2x^2 + x - 7 = 0$$ 5. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=2$, $b=1$, $c=-7$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times 2 \times (-7) = 1 + 56 = 57$$ 7. **Find the roots:** $$x = \frac{-1 \pm \sqrt{57}}{2 \times 2} = \frac{-1 \pm \sqrt{57}}{4}$$ 8. **Evaluate the square root and solutions:** $$\sqrt{57} \approx 7.550$$ - First root: $$x_1 = \frac{-1 + 7.550}{4} = \frac{6.550}{4} = 1.6375 \approx 1.64$$ - Second root: $$x_2 = \frac{-1 - 7.550}{4} = \frac{-8.550}{4} = -2.1375 \approx -2.14$$ **Final answers:** $$x \approx 1.64 \text{ or } x \approx -2.14$$