1. **State the problem:** We have the quadratic equation $$k + 9 = (6 - x)(2x + 1)$$ and need to find: (a) The range of values of $$k$$ for which the equation has two distinct real roots. (b) For the largest integer $$k$$ in that range, solve the quadratic equation. 2. **Expand and rearrange the equation:** Expand the right side: $$(6 - x)(2x + 1) = 6 \times 2x + 6 \times 1 - x \times 2x - x \times 1 = 12x + 6 - 2x^2 - x = -2x^2 + 11x + 6$$ So the equation becomes: $$k + 9 = -2x^2 + 11x + 6$$ Rearranged to standard quadratic form: $$-2x^2 + 11x + 6 - (k + 9) = 0$$ $$-2x^2 + 11x + 6 - k - 9 = 0$$ $$-2x^2 + 11x - (k + 3) = 0$$ 3. **Rewrite the quadratic equation:** $$-2x^2 + 11x - (k + 3) = 0$$ Multiply both sides by $$-1$$ to simplify: $$2x^2 - 11x + (k + 3) = 0$$ 4. **Condition for two distinct real roots:** For a quadratic $$ax^2 + bx + c = 0$$ to have two distinct real roots, the discriminant $$\Delta$$ must be positive: $$\Delta = b^2 - 4ac > 0$$ Here, $$a = 2$$, $$b = -11$$, $$c = k + 3$$. Calculate discriminant: $$\Delta = (-11)^2 - 4 \times 2 \times (k + 3) = 121 - 8(k + 3)$$ 5. **Find range of $$k$$:** Set $$\Delta > 0$$: $$121 - 8(k + 3) > 0$$ $$121 - 8k - 24 > 0$$ $$97 - 8k > 0$$ $$-8k > -97$$ Divide both sides by $$-8$$, reversing inequality: $$\cancel{-8}k > \cancel{-97}$$ $$\cancel{-8}k > \cancel{-97}$$ $$k < \frac{97}{8}$$ So the range of $$k$$ for two distinct real roots is: $$k < 12.125$$ 6. **Find the largest integer $$k$$:** The largest integer less than $$12.125$$ is $$12$$. 7. **Solve the quadratic for $$k = 12$$:** Substitute $$k = 12$$ into the quadratic: $$2x^2 - 11x + (12 + 3) = 0$$ $$2x^2 - 11x + 15 = 0$$ Calculate discriminant: $$\Delta = (-11)^2 - 4 \times 2 \times 15 = 121 - 120 = 1$$ Use quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{11 \pm 1}{4}$$ Calculate roots: $$x_1 = \frac{11 + 1}{4} = \frac{12}{4} = 3$$ $$x_2 = \frac{11 - 1}{4} = \frac{10}{4} = 2.5$$ **Final answers:** (a) The range of $$k$$ is $$k < 12.125$$. (b) For $$k = 12$$, the roots are $$x = 3$$ and $$x = 2.5$$.