Question

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) 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 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. There will be one real solution since the discriminant is negative. There will be no real solutions since the discriminant is negative. 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 first need to calculate the discriminant. The discriminant \(D\) 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:

- \(a = 4\)
- \(b = 3\)
- \(c = -7\)

Now, plug 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, it indicates that there are two distinct real solutions.

Therefore, the correct statement is:

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