Simultaneous Equations 625046

1. **State the problem:** We need to find the points where the graphs of $$y = x^2 - 5x - 3$$ and $$y = x - 4$$ intersect. 2. **Set the equations equal:** Since both equal $$y$$, set them equal to find $$x$$: $$x^2 - 5x - 3 = x - 4$$ 3. **Rearrange to form a quadratic equation:** $$x^2 - 5x - 3 - x + 4 = 0$$ $$x^2 - 6x + 1 = 0$$ 4. **Use the quadratic formula:** For $$ax^2 + bx + c = 0$$, $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. Here, $$a=1$$, $$b=-6$$, $$c=1$$. $$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \times 1 \times 1}}{2 \times 1} = \frac{6 \pm \sqrt{36 - 4}}{2} = \frac{6 \pm \sqrt{32}}{2}$$ 5. **Simplify the square root:** $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$$ So, $$x = \frac{6 \pm 4\sqrt{2}}{2} = 3 \pm 2\sqrt{2}$$ 6. **Calculate approximate values:** $$2\sqrt{2} \approx 2 \times 1.414 = 2.828$$ So, $$x_1 = 3 + 2.828 = 5.828$$ $$x_2 = 3 - 2.828 = 0.172$$ 7. **Find corresponding $$y$$ values using $$y = x - 4$$:** For $$x_1 = 5.828$$: $$y_1 = 5.828 - 4 = 1.828$$ For $$x_2 = 0.172$$: $$y_2 = 0.172 - 4 = -3.828$$ 8. **Final answers:** $$x \approx 5.83, y \approx 1.83$$ or $$x \approx 0.17, y \approx -3.83$$