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

Answer 1

First, we need to calculate the discriminant of the quadratic equation \(4x^2 + 3x - 7 = 0\). The discriminant (\(D\)) can be calculated using the formula:

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

where \(a = 4\), \(b = 3\), and \(c = -7\).

Now we plug in the values:

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

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

Therefore, the correct statement is:

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