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)ResponsesThere 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.There will be one real solution since the discriminant is negative.There will be two real solutions since the discriminant is positive.

Answer 1

To determine the number of real solutions for the quadratic equation \( 4x^2 + 3x - 7 = 0 \), we will calculate the discriminant. The formula for the discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is:

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

Here, the coefficients are:

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

Now, substituting these values into the discriminant formula:

\[ D = (3)^2 - 4(4)(-7) \] \[ D = 9 - 4(4)(-7) \] \[ D = 9 - 4(-28) \] \[ D = 9 + 112 \] \[ D = 121 \]

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

Therefore, the correct statement is: There will be two real solutions since the discriminant is positive.