Question

9. Determine whether (Image: start root a end root plus start root a end root equals 2 a) is sometimes, always, or never true. (1 point)

never
sometimes
always

Answer 1

[AI-generated answer — Bot]

always

Answer 2

AAAaannndd the bot gets it wrong yet again!

√a + √a = 2
sometimes true
only is a=1

Answer 3

[AI-generated answer — Bot]

I apologize for the mistake in my previous response. You are correct that the statement (Image: start root a end root plus start root a end root equals 2 a) is sometimes true only when a = 1. Thank you for bringing this to my attention.

Answer 4

[AI-generated answer — Explain Bot]

To determine whether the equation \( \sqrt{a} + \sqrt{a} = 2a \) is sometimes, always, or never true, we can use algebraic methods.

Let's start by simplifying the equation. Since both terms on the left side of the equation are square roots of the same number "a," we can combine them as follows:

\( 2\sqrt{a} = 2a \)

Next, to isolate the square root term, we can divide both sides of the equation by 2:

\( \sqrt{a} = a \)

Now, we need to square both sides of the equation to eliminate the square root:

\( (\sqrt{a})^2 = a^2 \)

Simplifying further:

\( a = a^2 \)

This equation tells us that the value of "a" must satisfy the condition \( a = a^2 \).

Here, we can notice that if "a" is equal to 0 or 1, the equation holds true:

For \( a = 0 \):
\( 0 = 0^2 = 0 \) (The equation is true)

For \( a = 1 \):
\( 1 = 1^2 = 1 \) (The equation is true)

However, for all other values of "a," the equation is not true:

For \( a = 2 \):
\( 2 \neq 2^2 = 4 \) (The equation is not true)

For \( a = 3 \):
\( 3 \neq 3^2 = 9 \) (The equation is not true)

We can continue this pattern for any other value of "a," and we will find that the equation is not true.

Therefore, the equation \( \sqrt{a} + \sqrt{a} = 2a \) is sometimes true (only when \( a = 0 \) or \( a = 1 \)), but it is not always true for all values of "a." Hence, the correct answer is "sometimes."