1. **Problem Statement:** Determine how many roots each quadratic equation has. 2. **Recall the quadratic formula and discriminant:** For a quadratic equation $ax^2 + bx + c = 0$, the roots are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The discriminant $\Delta = b^2 - 4ac$ tells us the number of roots: - If $\Delta > 0$, there are 2 distinct real roots. - If $\Delta = 0$, there is 1 real root (a repeated root). - If $\Delta < 0$, there are no real roots (complex roots). 3. **Equation 1: $y = x^2 + 3x - 8$** - Here, $a=1$, $b=3$, $c=-8$. - Calculate discriminant: $$\Delta = 3^2 - 4 \times 1 \times (-8) = 9 + 32 = 41$$ - Since $41 > 0$, there are 2 distinct real roots. 4. **Equation 2: $y = 0.25x^2 + 2$** - Here, $a=0.25$, $b=0$, $c=2$. - Calculate discriminant: $$\Delta = 0^2 - 4 \times 0.25 \times 2 = 0 - 2 = -2$$ - Since $-2 < 0$, there are no real roots. 5. **Equation 3: $y = -2x^2 + 8x - 8$** - Here, $a=-2$, $b=8$, $c=-8$. - Calculate discriminant: $$\Delta = 8^2 - 4 \times (-2) \times (-8) = 64 - 64 = 0$$ - Since $\Delta = 0$, there is exactly 1 real root (a repeated root). **Final answers:** - Equation 1 has 2 real roots. - Equation 2 has 0 real roots. - Equation 3 has 1 real root.