1. **State the problem:** Solve the simultaneous equations: $$y = 2x^2 - 7$$ $$y = 3x + 20$$ 2. **Set the equations equal to each other** since both equal $y$: $$2x^2 - 7 = 3x + 20$$ 3. **Rearrange the equation to standard quadratic form:** $$2x^2 - 3x - 27 = 0$$ 4. **Use the quadratic formula** to solve for $x$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-3$, and $c=-27$. 5. **Calculate the discriminant:** $$b^2 - 4ac = (-3)^2 - 4 \times 2 \times (-27) = 9 + 216 = 225$$ 6. **Find the square root of the discriminant:** $$\sqrt{225} = 15$$ 7. **Calculate the two possible values for $x$:** $$x = \frac{-(-3) \pm 15}{2 \times 2} = \frac{3 \pm 15}{4}$$ 8. **First solution:** $$x = \frac{3 + 15}{4} = \frac{18}{4} = \frac{9}{2} = 4.5$$ 9. **Second solution:** $$x = \frac{3 - 15}{4} = \frac{-12}{4} = -3$$ 10. **Find corresponding $y$ values by substituting back into $y = 3x + 20$:** For $x=4.5$: $$y = 3(4.5) + 20 = 13.5 + 20 = 33.5$$ For $x=-3$: $$y = 3(-3) + 20 = -9 + 20 = 11$$ **Final solutions:** $$(x, y) = \left(4.5, 33.5\right) \text{ and } (-3, 11)$$