Question

what does it mean when the discriminant of a quadratic equation is zero?

a. the equation has no solution

b. the equation has one real solution

c. the equation has two distinct real solutions

d. the equation has two complex solutions

Explanation:

For a quadratic equation \( ax^{2}+bx + c = 0\) (\(a
eq0\)), the discriminant is given by \( D=b^{2}-4ac\). The nature of the roots is determined by the discriminant:

• If \( D>0\), the equation has two distinct real solutions.
• If \( D = 0\), the equation has one real solution (a repeated root, or a root with multiplicity 2).
• If \( D<0\), the equation has two complex conjugate solutions.

Now let's analyze each option:

• Option a: If \( D = 0\), the equation has a solution (a repeated real root), so this is incorrect.
• Option b: When \( D=0\), the quadratic formula \(x=\frac{-b\pm\sqrt{D}}{2a}\) becomes \(x = \frac{-b}{2a}\) (since \(\sqrt{0}=0\)), so there is one real solution (a repeated root). This is correct.
• Option c: Two distinct real solutions occur when \( D>0\), not when \( D = 0\), so this is incorrect.
• Option d: Two complex solutions occur when \( D<0\), not when \( D = 0\), so this is incorrect.

Answer:

b. The equation has one real solution