Exam Questions 7C6D7A

1. **Problem:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula:** Use the factorization method or quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-5$$, and $$c=6$$. 3. **Step-by-step solution:** 1. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$. 2. Since $$\Delta > 0$$, there are two real roots. 3. Apply the quadratic formula: $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$. 4. Calculate each root: $$x_1 = \frac{5 + 1}{2} = \frac{6}{2} = 3$$ $$x_2 = \frac{5 - 1}{2} = \frac{4}{2} = 2$$ 4. **Answer:** The solutions are $$x=3$$ and $$x=2$$. --- 1. **Problem:** Simplify the expression $$\frac{2x^2 - 8}{4x}$$. 2. **Formula:** Simplify by factoring numerator and canceling common factors. 3. **Step-by-step solution:** 1. Factor numerator: $$2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)$$. 2. Write the expression: $$\frac{2(x - 2)(x + 2)}{4x}$$. 3. Simplify the fraction $$\frac{2}{4} = \frac{\cancel{2}}{2 \times \cancel{2}} = \frac{1}{2}$$: $$\frac{\cancel{2}(x - 2)(x + 2)}{2 \times \cancel{2} x} = \frac{(x - 2)(x + 2)}{2x}$$. 4. **Answer:** The simplified expression is $$\frac{(x - 2)(x + 2)}{2x}$$. --- 1. **Problem:** Find the slope and intercepts of the line $$y = 3x - 4$$. 2. **Formula:** The slope-intercept form is $$y = mx + c$$ where $$m$$ is slope and $$c$$ is y-intercept. 3. **Step-by-step solution:** 1. Identify slope $$m = 3$$. 2. Identify y-intercept $$c = -4$$. 3. Find x-intercept by setting $$y=0$$: $$0 = 3x - 4 \Rightarrow 3x = 4 \Rightarrow x = \frac{4}{3}$$. 4. **Answer:** Slope is $$3$$, y-intercept is $$-4$$, x-intercept is $$\frac{4}{3}$$.