Assuming an equation with one side as a squared variable expression and the other side as a numeric expression, which of the following statements is correct?

Question

GPT-4o mini · Answer

To provide you with the correct statements regarding an equation that has one side as a squared variable expression and the other side as a numeric expression, I would need to know the specific equation or the options you're referring to. However, I can offer some general statements that are commonly correct for such equations:

If the squared variable expression is equal to a positive numeric expression (e.g., $x^2 = k$, where $k > 0$), then there will be two real solutions $x = \sqrt{k}$ and $x = -\sqrt{k}$.
If the squared variable expression is equal to zero (e.g., $x^2 = 0$), then there is exactly one solution: $x = 0$.
If the squared variable expression is equal to a negative numeric expression (e.g., $x^2 = -k$, where $k > 0$), there are no real solutions, but there would be two complex solutions: $x = i\sqrt{k}$ and $x = -i\sqrt{k}$.
The principle of squaring both sides of the equation can lead to extraneous solutions, so it is essential to check any solutions found in the original equation.

If you have specific options provided, please share them for a more tailored response.

Answer

Statement #2 is correct.

GPT-4o mini · Answer

Yes, statement #2 is correct: **If the squared variable expression is equal to zero (e.g., $x^2 = 0$), then there is exactly one solution: $x = 0$.** This is because the only value for $x$ that satisfies $x^2 = 0$ is $x = 0$. If you have any further questions or additional statements you'd like to discuss, feel free to share!

Answer

Solve x2=78 by inspection. There are two real solutions. Enter the lesser number first, and use exponents in your responses. (1 point)

GPT-4o mini · Answer

To solve the equation $x^2 = 78$ by inspection, we can find the square roots of $78$. Calculating the square roots gives us: $$ x = \sqrt{78} \quad \text{and} \quad x = -\sqrt{78} $$ Now, we know that $\sqrt{78}$ is approximately $8.83$. Thus, the two real solutions are: 1. The lesser solution: $-\sqrt{78}$ 2. The greater solution: $\sqrt{78}$ So, the final answer with the lesser number first is: $$ -\sqrt{78}, \sqrt{78} $$

Answer

Is this correct

GPT-4o mini · Answer

Yes, that is correct! The equation $x^2 = 78$ indeed has two real solutions: 1. The lesser solution: $-\sqrt{78}$ 2. The greater solution: $\sqrt{78}$ In interval notation, you would express these as: $$ -\sqrt{78}, \sqrt{78} $$ So, you have correctly identified both solutions as $ -\sqrt{78} $ and $ \sqrt{78} $, with the lesser number listed first. If you have any further questions or need additional clarification, feel free to ask!

Answer

Solve the following quadratic equation using square roots: −9+(x−3)2−4=−4 .

GPT-4o mini · Answer

To solve the equation $$ -9 + (x - 3)^2 - 4 = -4, $$ let's first simplify the equation step by step. 1. First, simplify the left side: $$ -9 - 4 + (x - 3)^2 = -4. $$ This simplifies to: $$ -13 + (x - 3)^2 = -4. $$ 2. Next, add 13 to both sides to isolate the squared term: $$ (x - 3)^2 = -4 + 13, $$ which simplifies to: $$ (x - 3)^2 = 9. $$ 3. Now, take the square root of both sides: $$ x - 3 = \pm 3. $$ This gives us two equations to solve: $$ x - 3 = 3 \quad \text{and} \quad x - 3 = -3. $$ 4. Solve each equation: - For $x - 3 = 3$: $$ x = 3 + 3 = 6. $$ - For $x - 3 = -3$: $$ x = -3 + 3 = 0. $$ 5. **The solutions are**: $$ x = 6 \quad \text{and} \quad x = 0. $$ So the complete solution set is $x = 0$ and $x = 6$.

Answer

16,−10} left brace 16 comma negative 10 right brace {8,−2} left brace 8 comma negative 2 right brace {2,−8} left brace 2 comma negative 8 right brace {28,−22}

GPT-4o mini · Answer

It looks like you provided several pairs of numbers enclosed in curly braces, which could represent potential solutions to equations or distinct pairs of values.

Given our previous discussion regarding the equation you provided, the solutions we derived were $ x = 6 $ and $ x = 0 $. The pairs you mentioned do not relate to that solution.

However, if you are discussing or comparing solutions for a specific quadratic equation or looking for a relationship among the sets you listed, please clarify what you'd like to know.

If you're looking for which of those pairs might represent roots or solutions to a quadratic equation or any particular context, here's how to interpret them:

{16, -10}: This set includes positive and negative numbers, could be solutions to a quadratic that yields these values.
{8, -2}: This represents another potential pair of solutions.
{2, -8}: Also another distinct scenario.
{28, -22}: Another pair of solutions.

If these pairs are to be involved in a specific mathematical context, such as verifying solutions or being roots of polynomial equations, please provide that context or a specific question so I can assist you more effectively!