1. **State the problem:** Determine the number of solutions for the simultaneous equations formed by pairing the parabola equation $y=2x^2 + 3$ with each linear equation $y=2x+b$, where $b=4.5$, $2.5$, and $0.5$. 2. **Set equations equal:** For each pair, solve: $$2x^2 + 3 = 2x + b$$ 3. **Rearrange each into standard quadratic form:** $$2x^2 - 2x + (3 - b) = 0$$ 4. **Calculate the discriminant $ abla$ for each to determine the number of solutions:** $$ abla = (-2)^2 - 4 \times 2 \times (3 - b) = 4 - 8(3 - b) = 4 - 24 + 8b = 8b - 20$$ 5. **Analyse each case:** - For $b=4.5$: $$\nabla = 8 \times 4.5 - 20 = 36 - 20 = 16 > 0$$ Initially, this suggests 2 solutions, but the graph shows the green line tangent (1 solution). Let's check carefully. 6. **Verify discriminant sign for $b=4.5$:** $$\nabla = 8 \times 4.5 - 20 = 36 - 20 = 16 > 0$$ A positive discriminant means 2 distinct solutions normally, but to confirm the tangent, check if the parabola and line share a point with equal slopes. 7. **Check slopes for tangency:** Slope of parabola derivative: $$\frac{dy}{dx} = 4x$$ Slope of line = 2. For tangency, $$4x = 2 \Rightarrow x=0.5$$ Substitute $x=0.5$ into equations: - Parabola: $y=2(0.5)^2 + 3 = 2(0.25)+3=0.5+3=3.5$ - Line: $y=2(0.5) + 4.5=1+4.5=5.5$ They don't equal; since values differ, no tangency here. Let's reconsider the earlier interpretation. 8. **Re-examining the discriminant and graphical hints:** The user states the green line is tangent with 1 solution, orange line intersects twice (2 solutions), and purple line no solution. Using formula $\nabla = 8b-20$: - For 1 solution (tangent), discriminant $=0$: $$8b - 20 = 0 \Rightarrow b = \frac{20}{8} = 2.5$$ So actually, - $b=2.5$ corresponds to 1 solution (tangency) - $b=4.5$ gives $\nabla = 16 > 0$ (2 solutions) - $b=0.5$ gives $\nabla = 8(0.5) - 20 = 4 - 20 = -16 < 0$ (no solution) 9. **Conclusion:** - a) 1 solution for $y=2x^2+3$ and $y=2x+2.5$ - b) 2 solutions for $y=2x^2+3$ and $y=2x+4.5$ - c) 0 solutions for $y=2x^2+3$ and $y=2x+0.5$