Use the discriminant to determine the number of real solutions of 4x2+3x−7=0 . Which of the following statements gives the correct explanation?(1 point)

Question

Responses

There will be one real solution since the discriminant is negative.
There will be one real solution since the discriminant is negative.

There will be one real solution since the discriminant is zero.
There will be one real solution since the discriminant is zero.

There will be two real solutions since the discriminant is positive.
There will be two real solutions since the discriminant is positive.

There will be no real solutions since the discriminant is negative.

Answer 1

To determine the number of real solutions for the quadratic equation \(4x^2 + 3x - 7 = 0\), we start by calculating the discriminant, which is given by the formula:

\[ D = b^2 - 4ac \]

In this equation, \(a = 4\), \(b = 3\), and \(c = -7\). Plugging these values into the discriminant formula:

\[ D = (3)^2 - 4(4)(-7) \] \[ D = 9 - 4 \cdot 4 \cdot (-7) \] \[ D = 9 - (-112) \] \[ D = 9 + 112 \] \[ D = 121 \]

Since the discriminant \(D = 121\) is positive, this indicates that there are two real solutions.

Therefore, the correct response is:

There will be two real solutions since the discriminant is positive.