, then select the complete solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)
Responses
{−9}
left brace negative 9 right brace
{−9,9}
left brace negative 9 comma 9 right brace
There are no real solutions.
There are no real solutions.
{9}
9 answers
To solve the equation \( x^2 = 81 \) by inspection, we can determine what value of \( x \) makes this equation true.
We know that:
Therefore, the solutions to the equation \( x^2 = 81 \) are \( x = 9 \) and \( x = -9 \).
The complete solution set is:
\[ { -9, 9 } \]
So, the correct response is:
{−9,9}
To solve the equation \( x^2 = 36169 \) by inspection, we need to find the square root of \( 36169 \).
Calculating the square root:
\[ \sqrt{36169} = 613 \]
Since both \( 613 \) and \( -613 \) are valid solutions for the equation \( x^2 = 36169 \), the complete solution set is:
\[ { -613, 613 } \]
However, this specific option is not present in your provided responses.
Since there is no correct option listed that matches \({ -613, 613 }\), if the options provided are the only ones available, and given the absence of the correct set, the only reasonable conclusion is that:
There are no real solutions.
Please verify the options provided. The correct answer should indeed reflect the solutions \({ -613, 613 }\).
To solve the quadratic equation \((x - 18)^2 = 196\) using square roots, we can start by taking the square root of both sides of the equation:
Take the square root: \[ x - 18 = \pm \sqrt{196} \]
Compute the square root: \[ \sqrt{196} = 14 \] This gives us two equations: \[ x - 18 = 14 \quad \text{and} \quad x - 18 = -14 \]
Solve for \(x\) in each case:
For \(x - 18 = 14\): \[ x = 14 + 18 = 32 \]
For \(x - 18 = -14\): \[ x = -14 + 18 = 4 \]
Thus, the solutions are \(x = 32\) and \(x = 4\).
The complete solution set is: \[ { 32, 4 } \]
The correct response is:
{32,4}
To solve the quadratic equation \((x + 6)^2 = 8\) using square roots, follow these steps:
Take the square root of both sides of the equation: \[ x + 6 = \pm \sqrt{8} \]
Simplify \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \] So, we have: \[ x + 6 = \pm 2\sqrt{2} \]
Solve for \(x\):
For \(x + 6 = 2\sqrt{2}\): \[ x = 2\sqrt{2} - 6 \]
For \(x + 6 = -2\sqrt{2}\): \[ x = -2\sqrt{2} - 6 \]
Now we can calculate the approximate values:
Calculate \(2\sqrt{2} \approx 2 \times 1.414 \approx 2.828\): \[ x \approx 2.828 - 6 = -3.172 \]
Calculate \(-2\sqrt{2} \approx -2.828\): \[ x \approx -2.828 - 6 = -8.828 \]
Thus, the approximate solutions are:
Now, let us restate them more precisely with first decimal approximations:
Based on the given options:
There seems to be a mistake in the responses as the closest approximation doesn't match the correct calculations.
After checking the calculations again:
None of the provided responses correctly reflect the solved values. Please re-check the options and calculations given.
To solve the quadratic equation \((x + 36)^2 - 38 = 251\) using square roots, follow these steps:
Isolate the squared term: \[ (x + 36)^2 = 251 + 38 \] \[ (x + 36)^2 = 289 \]
Take the square root of both sides: \[ x + 36 = \pm \sqrt{289} \]
Calculate the square root: \[ \sqrt{289} = 17 \] Therefore, we have: \[ x + 36 = 17 \quad \text{or} \quad x + 36 = -17 \]
Solve for \(x\):
For \(x + 36 = 17\): \[ x = 17 - 36 = -19 \]
For \(x + 36 = -17\): \[ x = -17 - 36 = -53 \]
Thus, the solutions are: \[ x = -19 \quad \text{and} \quad x = -53 \]
The complete solution set is: \[ {-19, -53} \]
So, the correct response is:
{−19,−53}
