1. The problem asks to find which quadratic equation has a discriminant equal to 0. 2. Recall the discriminant formula for a quadratic equation $ax^2 + bx + c = 0$ is: $$\Delta = b^2 - 4ac$$ 3. The discriminant tells us about the nature of the roots: - If $\Delta > 0$, two distinct real roots. - If $\Delta = 0$, exactly one real root (a repeated root). - If $\Delta < 0$, no real roots (complex roots). 4. Calculate the discriminant for each equation: - For $x^2 - 5x + 6 = 0$, $a=1$, $b=-5$, $c=6$: $$\Delta = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ - For $x^2 - x + 2 = 0$, $a=1$, $b=-1$, $c=2$: $$\Delta = (-1)^2 - 4 \times 1 \times 2 = 1 - 8 = -7$$ - For $x^2 + 5x - 2 = 0$, $a=1$, $b=5$, $c=-2$: $$\Delta = 5^2 - 4 \times 1 \times (-2) = 25 + 8 = 33$$ - For $x^2 - 12x + 36 = 0$, $a=1$, $b=-12$, $c=36$: $$\Delta = (-12)^2 - 4 \times 1 \times 36 = 144 - 144 = 0$$ - For $x^2 - 5x = 0$, rewrite as $x^2 - 5x + 0 = 0$, $a=1$, $b=-5$, $c=0$: $$\Delta = (-5)^2 - 4 \times 1 \times 0 = 25 - 0 = 25$$ 5. Only the equation $x^2 - 12x + 36 = 0$ has discriminant equal to 0. **Final answer:** $x^2 - 12x + 36 = 0$ has discriminant 0.