Final answer:
The nature of the roots of a quadratic equation can be determined by looking at the discriminant, which is the expression inside the square root of the quadratic formula. In the given quadratic equation 4x²-12x + 9=0, the discriminant is zero, indicating that the quadratic equation has one real root.
Step-by-step explanation:
The nature of the roots of a quadratic equation can be determined by looking at the discriminant, which is the expression inside the square root of the quadratic formula. The discriminant is given by b^2 - 4ac. If the discriminant is positive, the quadratic equation will have two distinct real roots. If the discriminant is zero, the quadratic equation will have one real root (a repeated root). And if the discriminant is negative, the quadratic equation will have two complex roots (conjugate pairs).
In the given quadratic equation 4x²-12x + 9=0, the coefficients are a = 4, b = -12, and c = 9. Calculating the discriminant using the formula b^2 - 4ac, we get (-12)^2 - 4(4)(9) = 144 - 144 = 0. Since the discriminant is zero, the quadratic equation has one real root.
So, the nature of the roots of the given quadratic equation is that it has one real root.