To find the discriminant and describe the nature of the roots for each quadratic equation, we use the formula for the discriminant $\Delta = b^2 - 4ac$, where the quadratic equation is in the form $ax^2 + bx + c = 0$. The nature of the roots is determined as follows: If $\Delta > 0$, there are two distinct real roots. If $\Delta = 0$, there is one real root (a repeated root). If $\Delta < 0$, there are two complex conjugate roots (no real roots). --- a. $x^2 - 4x + 3 = 0$ Step 1: Identify $a, b, c$. Here, $a = 1$, $b = -4$, $c = 3$. Step 2: Calculate the discriminant. $$ \Delta = b^2 - 4ac = (-4)^2 - 4(1)(3) $$ $$ \Delta = 16 - 12 $$ $$ \Delta = 4 $$ Step 3: Describe the nature of the roots. Since $\Delta = 4 > 0$, the equation has two distinct real roots. The discriminant is $\boxed{4}$. --- b. $x^2 - 6x + 9 = 0$ Step 1: Identify $a, b, c$. Here, $a = 1$, $b = -6$, $c = 9$. Step 2: Calculate the discriminant. $$ \Delta = b^2 - 4ac = (-6)^2 - 4(1)(9) $$ $$ \Delta = 36 - 36 $$ $$ \Delta = 0 $$ Step 3: Describe the nature of the roots. Since $\Delta = 0$, the equation has one real root (a repeated root). The discriminant is $\boxed{0}$. --- c. $x^2 + 2x + 5 = 0$ Step 1: Identify $a, b, c$. Here, $a = 1$, $b = 2$, $c = 5$. Step 2: Calculate the discriminant. $$ \Delta = b^2 - 4ac = (2)^2 - 4(1)(5) $$ $$ \Delta = 4 - 20 $$ $$ \Delta = -16 $$ Step 3: Describe the nature of the roots. Since $\Delta = -16 < 0$, the equation has two complex conjugate roots (no real roots). The discriminant is $\boxed{-16}$. --- d. $3x^2 - 12x + 12 = 0$ Step 1: Identify $a, b, c$. Here, $a = 3$, $b = -12$, $c = 12$. Step 2: Calculate the discriminant. $$ \Delta = b^2 - 4ac = (-12)^2 - 4(3)(12) $$ $$ \Delta = 144 - 144 $$ $$ \Delta = 0 $$ Step 3: Describe the nature of the roots. Since $\Delta = 0$, the equation has one real root (a repeated root). The discriminant is $\boxed{0}$. --- e. $x^2 - 7x + 10 = 0$ Step 1: Identify $a, b, c$. Here, $a = 1$, $b = -7$, $c = 10$. Step 2: Calculate the discriminant. $$ \Delta = b^2 - 4ac = (-7)^2 - 4(1)(10) $$ $$ \Delta = 49 - 40 $$ $$ \Delta = 9 $$ Step 3: Describe the nature of the roots. Since $\Delta = 9 > 0$, the equation has two distinct real roots. The discriminant is $\boxed{9}$.