1. **State the problem:** Solve the quadratic equation $$x^2 - 12x + 7 = 0$$ and find the value of $$jk$$ where one solution is written as $$j - \sqrt{k}$$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $$a=1$$, $$b=-12$$, and $$c=7$$. 3. **Calculate the discriminant:** $$ \Delta = b^2 - 4ac = (-12)^2 - 4 \times 1 \times 7 = 144 - 28 = 116 $$ 4. **Write the solutions:** $$ x = \frac{-(-12) \pm \sqrt{116}}{2 \times 1} = \frac{12 \pm \sqrt{116}}{2} $$ 5. **Simplify the square root:** $$ \sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29} $$ 6. **Substitute back:** $$ x = \frac{12 \pm 2\sqrt{29}}{2} $$ 7. **Cancel common factor 2:** $$ x = \frac{\cancel{2} \times 6 \pm \cancel{2} \times \sqrt{29}}{\cancel{2}} = 6 \pm \sqrt{29} $$ 8. **Identify constants:** One solution is $$j - \sqrt{k}$$, so $$j=6$$ and $$k=29$$. 9. **Calculate $$jk$$:** $$ jk = 6 \times 29 = 174 $$ **Final answer:** $$jk = 174$$