1. **Problem:** Solve the quadratic equation $$2x^2 - 5x + 3 = 0$$. 2. **Step 1:** Identify coefficients: $$a=2$$, $$b=-5$$, $$c=3$$. 3. **Step 2:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 4. **Step 3:** Calculate the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times 3 = 25 - 24 = 1$$. 5. **Step 4:** Substitute values into the formula: $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 2} = \frac{5 \pm 1}{4}$$. 6. **Step 5:** Find the two solutions: $$x_1 = \frac{5 + 1}{4} = \frac{6}{4} = 1.5$$ $$x_2 = \frac{5 - 1}{4} = \frac{4}{4} = 1$$. 7. **Answer:** The solutions are $$x = 1.5$$ and $$x = 1$$. --- 1. **Problem:** Simplify the expression $$\frac{3x^2 - 12}{6x}$$. 2. **Step 1:** Factor numerator: $$3x^2 - 12 = 3(x^2 - 4)$$. 3. **Step 2:** Recognize difference of squares: $$x^2 - 4 = (x - 2)(x + 2)$$. 4. **Step 3:** Rewrite expression: $$\frac{3(x - 2)(x + 2)}{6x}$$. 5. **Step 4:** Simplify coefficients: $$\frac{3}{6} = \frac{1}{2}$$. 6. **Step 5:** Final simplified form: $$\frac{(x - 2)(x + 2)}{2x}$$. 7. **Answer:** $$\frac{(x - 2)(x + 2)}{2x}$$. --- 1. **Problem:** Find the equation of the line passing through points $$(1, 2)$$ and $$(3, 8)$$. 2. **Step 1:** Calculate the gradient $$m$$: $$m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$$. 3. **Step 2:** Use point-slope form: $$y - y_1 = m(x - x_1)$$ with point $$(1, 2)$$: $$y - 2 = 3(x - 1)$$. 4. **Step 3:** Expand and simplify: $$y - 2 = 3x - 3$$ $$y = 3x - 1$$. 5. **Answer:** The equation of the line is $$y = 3x - 1$$.