1. **Problem Statement:** We are given the function $$y = x(x - 4) - 5$$ which simplifies to $$y = x^2 - 4x - 5$$. We need to draw its graph for $$-3 \leq x \leq 4$$ and use it to solve the following equations: a) $$x^2 - 5x - 3 = 0$$ b) $$x^2 - 3x - 4 = 0$$ 2. **Graphing the function:** The function is a quadratic polynomial, a parabola opening upwards because the coefficient of $$x^2$$ is positive. The vertex formula for a parabola $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $$a = 1$$ and $$b = -4$$, so $$x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2$$. The vertex point is at $$x=2$$. Calculate $$y$$ at $$x=2$$: $$y = 2^2 - 4 \times 2 - 5 = 4 - 8 - 5 = -9$$. So the vertex is at $$(2, -9)$$. 3. **Using the graph to solve the equations:** We want to find the roots of the given equations by comparing them to the graph of $$y = x^2 - 4x - 5$$. Rewrite each equation in the form $$y = 0$$: For a) $$x^2 - 5x - 3 = 0$$ For b) $$x^2 - 3x - 4 = 0$$ 4. **Relate each equation to the graph:** Note that the graph is $$y = x^2 - 4x - 5$$. We can write each equation as: For a): $$x^2 - 5x - 3 = 0$$ Rewrite as $$x^2 - 4x - 5 = (x^2 - 5x - 3) + (x - 2)$$ So, $$y = x^2 - 4x - 5 = (x^2 - 5x - 3) + (x - 2)$$ Set $$y = 0$$ to find roots of the original equation: $$0 = (x^2 - 5x - 3) + (x - 2)$$ This means the roots of $$x^2 - 5x - 3 = 0$$ are where the graph $$y = x^2 - 4x - 5$$ intersects the line $$y = -(x - 2)$$. Similarly for b): $$x^2 - 3x - 4 = 0$$ Rewrite as $$x^2 - 4x - 5 = (x^2 - 3x - 4) - (x + 1)$$ Set $$y = 0$$: $$0 = (x^2 - 3x - 4) - (x + 1)$$ So roots of $$x^2 - 3x - 4 = 0$$ are where the graph $$y = x^2 - 4x - 5$$ intersects the line $$y = x + 1$$. 5. **Solve each quadratic directly (for exact roots):** For a) $$x^2 - 5x - 3 = 0$$ Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{5 \pm \sqrt{25 + 12}}{2} = \frac{5 \pm \sqrt{37}}{2}$$ Approximate roots: $$x \approx \frac{5 + 6.08}{2} = 5.54$$ (outside domain) and $$x \approx \frac{5 - 6.08}{2} = -0.54$$ (inside domain). For b) $$x^2 - 3x - 4 = 0$$ Quadratic formula: $$x = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2}$$ Roots: $$x = 4$$ and $$x = -1$$ (both inside domain). 6. **Summary:** - The graph of $$y = x^2 - 4x - 5$$ is a parabola with vertex at $$(2, -9)$$. - Equation a) has one root in the domain at approximately $$x = -0.54$$. - Equation b) has two roots in the domain at $$x = -1$$ and $$x = 4$$. These roots correspond to the points where the graph intersects the x-axis for the respective equations.