Question

mattie uses the discriminant to determine the number of zeros the quadratic equation $0 = 3x^2 - 7x + 4$ has. which best describes the discriminant and the number of zeros?

• the equation has one zero because the discriminant is 1.
• the equation has one zero because the discriminant is a perfect square.
• the equation has two zeros because the discriminant is greater than 0.
• the equation has no zeros because the discriminant is not a perfect square.

Explanation:

Step1: Recall discriminant formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the discriminant \(D=b^{2}-4ac\). Here, \(a = 3\), \(b=-7\), \(c = 4\).

Step2: Calculate discriminant

Substitute values into the formula: \(D=(-7)^{2}-4\times3\times4=49 - 48 = 1\).

Step3: Analyze discriminant and zeros

Since \(D = 1>0\), a quadratic equation with a positive discriminant has two distinct real zeros. Also, \(D = 1\) is a perfect square, but the key here is the sign of the discriminant for the number of zeros. However, let's check the options:

• First option: Says one zero, but \(D = 1>0\) means two zeros, so wrong.
• Second option: Says one zero, but \(D>0\) means two zeros, wrong.
• Third option: \(D = 1>0\), so two zeros, correct.
• Fourth option: \(D = 1\) is a perfect square and \(D>0\) means two zeros, so wrong.

Answer:

The equation has two zeros because the discriminant is greater than 0. (The corresponding option among the given choices, likely the third option as per the analysis)