Question

anderson uses the discriminant to correctly find the number of real solutions of the quadratic equation (\frac{1}{2}x^2 + 4x + 8 = 0). which explanation could anderson provide?

• the equation has no real number solutions because the discriminant is 0.
• the equation has one real number solution because the discriminant is 0.
• the equation has no real number solutions because the discriminant is less than 0.
• the equation has two real number solutions because the discriminant is greater than 0.

Explanation:

Step1: Recall discriminant formula

For quadratic equation \( ax^2 + bx + c = 0 \), discriminant \( D = b^2 - 4ac \).
Given equation: \( \frac{1}{2}x^2 + 4x + 8 = 0 \), so \( a=\frac{1}{2} \), \( b = 4 \), \( c = 8 \).

Step2: Calculate discriminant

\( D=(4)^2 - 4\times\frac{1}{2}\times8 \)
\( = 16 - 16 \)
\( = 0 \)

Step3: Relate discriminant to solutions

If \( D = 0 \), quadratic has one real solution (a repeated root).
Now check options:

• Option 1: Wrong (D=0 means one real solution, not no solution).
• Option 2: Correct (D=0, so one real solution).
• Option 3: Wrong (D=0, not less than 0).
• Option 4: Wrong (D=0, not greater than 0).

Answer:

The equation has one real number solution because the discriminant is 0.