To determine which equation has only one distinct solution, we need to look for an equation where the discriminant is zero. The discriminant \( D \) for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
If the discriminant \( D = 0 \), the equation has one distinct solution (also called a repeated or double root).
Let's calculate the discriminant for each option:
a. \( x^2 + 4x + 4 = 0 \)
- Here, \( a = 1 \), \( b = 4 \), \( c = 4 \)
- Discriminant: \( D = 4^2 - 4 \times 1 \times 4 = 16 - 16 = 0 \)
b. \( x^2 + 5x + 4 = 0 \)
- Here, \( a = 1 \), \( b = 5 \), \( c = 4 \)
- Discriminant: \( D = 5^2 - 4 \times 1 \times 4 = 25 - 16 = 9 \)
c. \( x^2 - 4 = 0 \)
- Here, \( a = 1 \), \( b = 0 \), \( c = -4 \)
- Discriminant: \( D = 0^2 - 4 \times 1 \times (-4) = 0 + 16 = 16 \)
d. \( x^2 - 5x + 4 = 0 \)
- Here, \( a = 1 \), \( b = -5 \), \( c = 4 \)
- Discriminant: \( D = (-5)^2 - 4 \times 1 \times 4 = 25 - 16 = 9 \)
Now, summarizing the results:
- a: \( D = 0 \) (one distinct solution)
- b: \( D = 9 \) (two distinct solutions)
- c: \( D = 16 \) (two distinct solutions)
- d: \( D = 9 \) (two distinct solutions)
Thus, the equation that has only one distinct solution is:
a. \( x^2 + 4x + 4 = 0 \)
1 answer