No Real Solution 5967Cb

1. **State the problem:** We need to find which quadratic equation among the four given has no real solution. 2. **Recall the discriminant formula:** For a quadratic equation $ax^2 + bx + c = 0$, the discriminant $\Delta$ is given by: $$\Delta = b^2 - 4ac$$ 3. **Important rule:** - If $\Delta > 0$, the equation has two distinct real solutions. - If $\Delta = 0$, the equation has exactly one real solution (a repeated root). - If $\Delta < 0$, the equation has no real solutions (complex roots). 4. **Calculate the discriminant for each equation:** - For $25x^2 - 10x + 4 = 0$: $$a=25, b=-10, c=4$$ $$\Delta = (-10)^2 - 4 \times 25 \times 4 = 100 - 400 = -300$$ - For $25x^2 + 10x - 4 = 0$: $$a=25, b=10, c=-4$$ $$\Delta = 10^2 - 4 \times 25 \times (-4) = 100 + 400 = 500$$ - For $25x^2 - 20x + 4 = 0$: $$a=25, b=-20, c=4$$ $$\Delta = (-20)^2 - 4 \times 25 \times 4 = 400 - 400 = 0$$ - For $25x^2 + 20x - 4 = 0$: $$a=25, b=20, c=-4$$ $$\Delta = 20^2 - 4 \times 25 \times (-4) = 400 + 400 = 800$$ 5. **Analyze the discriminants:** - Equation 1: $\Delta = -300 < 0$ (no real solutions) - Equation 2: $\Delta = 500 > 0$ (two real solutions) - Equation 3: $\Delta = 0$ (one real solution) - Equation 4: $\Delta = 800 > 0$ (two real solutions) 6. **Conclusion:** The quadratic equation with no real solution is: $$25x^2 - 10x + 4 = 0$$