Question

which equation has only one distinct solution?
a x^2 + 4x + 4 = 0
b x^2 + 5x + 4 = 0
c x^2 - 4 = 0
d x^2 - 5x + 4 = 0

Answer 1

To determine which equation has only one distinct solution, we can check the discriminant of each quadratic equation. The discriminant (\(D\)) is given by the formula:

\[
D = b^2 - 4ac
\]

A quadratic equation has only one distinct solution when the discriminant is equal to 0 (\(D = 0\)).

Let's evaluate each equation:

**a) \(x^2 + 4x + 4 = 0\)**

Here, \(a = 1\), \(b = 4\), and \(c = 4\).

\[
D = 4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0
\]

This equation has only one distinct solution.

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**b) \(x^2 + 5x + 4 = 0\)**

Here, \(a = 1\), \(b = 5\), and \(c = 4\).

\[
D = 5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9
\]

This equation has two distinct solutions.

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**c) \(x^2 - 4 = 0\)**

Here, \(a = 1\), \(b = 0\), and \(c = -4\).

\[
D = 0^2 - 4 \cdot 1 \cdot (-4) = 0 + 16 = 16
\]

This equation also has two distinct solutions.

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**d) \(x^2 - 5x + 4 = 0\)**

Here, \(a = 1\), \(b = -5\), and \(c = 4\).

\[
D = (-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9
\]

This equation has two distinct solutions.

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The only equation that has one distinct solution is:

**a) \(x^2 + 4x + 4 = 0\)**.