Question: In the quadratic equation 3x^2 - 12x + 9 = 0, what is the nature of the roots?
Options:
Two distinct real roots
One real root
Two complex roots
No roots
Correct Answer: One real root
Solution:
The discriminant is zero (0), indicating one real root (a repeated root).
In the quadratic equation 3x^2 - 12x + 9 = 0, what is the nature of the roots?
Practice Questions
Q1
In the quadratic equation 3x^2 - 12x + 9 = 0, what is the nature of the roots?
Two distinct real roots
One real root
Two complex roots
No roots
Questions & Step-by-Step Solutions
In the quadratic equation 3x^2 - 12x + 9 = 0, what is the nature of the roots?
Step 1: Identify the coefficients in the quadratic equation 3x^2 - 12x + 9 = 0. Here, a = 3, b = -12, and c = 9.
Step 2: Calculate the discriminant using the formula D = b^2 - 4ac.
Step 3: Substitute the values of a, b, and c into the discriminant formula: D = (-12)^2 - 4(3)(9).
Step 4: Calculate (-12)^2, which is 144.
Step 5: Calculate 4(3)(9), which is 108.
Step 6: Subtract 108 from 144 to find the discriminant: D = 144 - 108 = 36.
Step 7: Determine the nature of the roots based on the value of the discriminant: If D > 0, there are two distinct real roots; if D = 0, there is one real root (a repeated root); if D < 0, there are no real roots.
Step 8: Since the discriminant D = 36, which is greater than 0, conclude that there are two distinct real roots.
Quadratic Equation – A polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
Discriminant – The value calculated as b^2 - 4ac, which determines the nature of the roots of a quadratic equation.
Nature of Roots – The classification of the roots based on the discriminant: two distinct real roots, one repeated real root, or two complex roots.
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