Two Solutions for All Quadratic Equations Quick Check

Question

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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.

Answer 1

To determine the number of real solutions for the quadratic equation \( 4x^2 + 3x - 7 = 0 \) using the discriminant, we first need to identify the coefficients \(a\), \(b\), and \(c\) from the equation in the standard form \( ax^2 + bx + c = 0 \):

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

The discriminant \(D\) is calculated using the formula:

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

Substituting in the values:

\[ D = 3^2 - 4(4)(-7) \]

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

Since the discriminant \(D\) is positive (\(121 > 0\)), this means that the quadratic equation has two distinct real solutions.

Therefore, the correct response is:

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