1. **State the problem:** We need to find the points of intersection A and B between the line $y = 5x + 6$ and the curve $xy = 8$. 2. **Write down the equations:** - Line: $y = 5x + 6$ - Curve: $xy = 8$ 3. **Substitute $y$ from the line into the curve equation:** $$x(5x + 6) = 8$$ 4. **Simplify the equation:** $$5x^2 + 6x = 8$$ 5. **Bring all terms to one side to form a quadratic equation:** $$5x^2 + 6x - 8 = 0$$ 6. **Solve the quadratic equation using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=5$, $b=6$, and $c=-8$. Calculate the discriminant: $$\Delta = 6^2 - 4 \times 5 \times (-8) = 36 + 160 = 196$$ Calculate the roots: $$x = \frac{-6 \pm \sqrt{196}}{2 \times 5} = \frac{-6 \pm 14}{10}$$ 7. **Find the two values of $x$:** - For $+$ sign: $$x = \frac{-6 + 14}{10} = \frac{8}{10} = 0.8$$ - For $-$ sign: $$x = \frac{-6 - 14}{10} = \frac{-20}{10} = -2$$ 8. **Find corresponding $y$ values using $y = 5x + 6$:** - When $x=0.8$: $$y = 5(0.8) + 6 = 4 + 6 = 10$$ - When $x=-2$: $$y = 5(-2) + 6 = -10 + 6 = -4$$ 9. **Write the coordinates of points A and B:** - $A = (0.8, 10)$ - $B = (-2, -4)$