Question

The discriminant of the quadratic equation \(ax^2 + x + a = 0\) is :

• A. \(\sqrt{1 - 4a^2}\)
• B. \(1 - 4a^2\)
• C. \(4a^2 - 1\)
• D. \(\sqrt{4a^2 - 1}\)

Hint:

Remember that the discriminant itself does not include the square root symbol. The square root is part of the quadratic formula (\(\sqrt{D}\)), but the discriminant is just the expression inside it.

Answer 1

The Correct Option is B

The problem requires us to find the discriminant of the quadratic equation \( ax^2 + x + a = 0 \). The discriminant, \( D \), of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\(D = b^2 - 4ac\)

For the given quadratic equation, the coefficients are:

• \(a_1 = a\)
• \(b = 1\)
• \(c = a\)

Substituting these coefficients into the discriminant formula, we get:

\(D = (1)^2 - 4(a)(a)\)

Calculating further:

• \(D = 1^2 - 4a^2\)
• \(D = 1 - 4a^2\)

Therefore, the discriminant of the given quadratic equation is \( 1 - 4a^2 \).

Now, let's examine the provided options to determine the correct answer:

• \(\sqrt{1 - 4a^2}\) - This is incorrect as discriminants are not expressed under square roots without specific context from the nature of roots.
• \(1 - 4a^2\) - This matches our computed discriminant.
• \(4a^2 - 1\) - This is simply the negative of our solution and thus incorrect in this context.
• \(\sqrt{4a^2 - 1}\) - The same reasoning as the first option; it incorrectly interprets the discriminant.

Hence, the correct answer is \(1 - 4a^2\).