Question

1. which statement best explains why the solutions are irrational to the quadratic equation

$2x^2 + x - 7 = 0$
?
a. the value of the discriminant is positive.
b. the value of the discriminant is equal to 0.
c. the value of the discriminant is negative.
d. the value of the discriminant is not a perfect square.

Explanation:

Step1: Recall discriminant formula

For quadratic equation \(ax^2 + bx + c = 0\), discriminant \(D = b^2 - 4ac\). Here, \(a = 2\), \(b = 1\), \(c = -7\).

Step2: Calculate discriminant

Substitute values: \(D = 1^2 - 4\times2\times(-7) = 1 + 56 = 57\).

Step3: Analyze discriminant properties

• \(D = 57>0\) (so A is true, C is false).
• \(D

eq0\) (so B is false).

• Check if 57 is a perfect square: \(7^2 = 49\), \(8^2 = 64\), so 57 is not a perfect square. The solutions of a quadratic are irrational if discriminant is positive but not a perfect square (since solutions are \(\frac{-b\pm\sqrt{D}}{2a}\); if \(D\) is non - perfect square positive, \(\sqrt{D}\) is irrational, making solutions irrational). So D explains why solutions are irrational.

Answer:

D. The value of the discriminant is not a perfect square.