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

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

Answer 1

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

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

where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

In this case, we have:

• \(a = 4\)
• \(b = 3\)
• \(c = -7\)

Now, let's calculate the discriminant:

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

Since the discriminant \(D\) is positive (\(121 > 0\)), this tells us that there are two distinct real solutions for the equation.

Thus, the correct statement is:

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