1. **State the problem:** Solve the quadratic equation $$2x^2 - 7x + 3 = 0$$ by factorisation. 2. **Recall the factorisation method:** For a quadratic equation $$ax^2 + bx + c = 0$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate product and sum:** Here, $$a = 2$$, $$b = -7$$, and $$c = 3$$. Calculate product: $$2 \times 3 = 6$$. We need two numbers that multiply to 6 and add to -7. 4. **Find the numbers:** The numbers are $$-6$$ and $$-1$$ because $$-6 \times -1 = 6$$ and $$-6 + (-1) = -7$$. 5. **Rewrite the middle term:** Rewrite $$-7x$$ as $$-6x - x$$: $$2x^2 - 6x - x + 3 = 0$$ 6. **Group terms:** $$(2x^2 - 6x) - (x - 3) = 0$$ 7. **Factor each group:** $$2x(x - 3) - 1(x - 3) = 0$$ 8. **Factor out common binomial:** $$(2x - 1)(x - 3) = 0$$ 9. **Solve each factor:** Set each factor equal to zero: $$2x - 1 = 0 \Rightarrow x = \frac{1}{2}$$ $$x - 3 = 0 \Rightarrow x = 3$$ **Final answer:** $$x = \frac{1}{2}$$ or $$x = 3$$.