Which equation has only one distinct solution?

Question

A: x^2 + 4x + 4 = 0

B: x^2 + 5x + 4 = 0

C: x^2 - 4 = 0

D: x^2 - 5x + 4 =0

GPT-4o mini · Answer

An equation has only one distinct solution when its discriminant is equal to zero. The discriminant for a quadratic equation of the form $a x^{2} + b x + c = 0$ is given by the formula $D = b^{2} - 4 a c$ .

Let's calculate the discriminant for each equation:

A: $x^{2} + 4 x + 4 = 0$

Here, $a = 1$ , $b = 4$ , $c = 4$ .
$D = 4^{2} - 4 \cdot 1 \cdot 4 = 16 - 16 = 0$ .
This equation has only one distinct solution.

B: $x^{2} + 5 x + 4 = 0$

Here, $a = 1$ , $b = 5$ , $c = 4$ .
$D = 5^{2} - 4 \cdot 1 \cdot 4 = 25 - 16 = 9$ .
This equation has two distinct solutions.

C: $x^{2} - 4 = 0$

This can be rewritten as $x^{2} + 0 x - 4 = 0$ , so $a = 1$ , $b = 0$ , $c = - 4$ .
$D = 0^{2} - 4 \cdot 1 \cdot (- 4) = 0 + 16 = 16$ .
This equation has two distinct solutions.

D: $x^{2} - 5 x + 4 = 0$

Here, $a = 1$ , $b = - 5$ , $c = 4$ .
$D = (- 5)^{2} - 4 \cdot 1 \cdot 4 = 25 - 16 = 9$ .
This equation has two distinct solutions.

In conclusion, the equation that has only one distinct solution is:

A: $x^{2} + 4 x + 4 = 0$ .