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Use inspection to solve the equation x^2=81 , then select the complete solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)

Responses

There are no real solutions.
There are no real solutions.

{9}
left brace 9 right brace

{−9,9}
left brace negative 9 comma 9 right brace

{−9}

9 answers

To solve the equation \( x^2 = 81 \) by inspection, we can look for the numbers whose square equals 81.

The two possible solutions for \( x \) are:

  1. \( x = 9 \) (since \( 9^2 = 81 \))
  2. \( x = -9 \) (since \( (-9)^2 = 81 \))

Thus, the complete solution set is \( {-9, 9} \).

So the correct response is: {−9, 9}

Use inspection to solve the equation x^2=36/169 , then select the correct solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)
Responses

{−6/13}
left brace negative Start Fraction 6 over 13 End Fraction right brace

{−6/13,6/13}
left brace negative Start Fraction 6 over 13 End Fraction comma Start Fraction 6 over 13 End Fraction right brace

There are no real solutions.
There are no real solutions.

{6/13}
left brace Start Fraction 6 over 13 End Fraction right brace

To solve the equation \( x^2 = \frac{36}{169} \) by inspection, we can find the square roots of both sides.

Taking the square root, we get: \[ x = \sqrt{\frac{36}{169}} \quad \text{or} \quad x = -\sqrt{\frac{36}{169}} \]

Calculating the square roots: \[ x = \frac{6}{13} \quad \text{and} \quad x = -\frac{6}{13} \]

Therefore, the complete solution set is: \[ \left{ -\frac{6}{13}, \frac{6}{13} \right} \]

So the correct response is: {−6/13, 6/13}.

Solve the following quadratic equation using square roots: (x−18)^2=196.(1 point)
Responses

{21,5}
left brace 21 comma 5 right brace

{32,4}
left brace 32 comma 4 right brace

​ {−4,−32}
​ left brace negative 4 comma negative 32 right brace

{214,−178}

To solve the quadratic equation \((x - 18)^2 = 196\) using square roots, we first take the square root of both sides:

\[ x - 18 = \sqrt{196} \quad \text{or} \quad x - 18 = -\sqrt{196} \]

Calculating the square root of 196 gives us:

\[ \sqrt{196} = 14 \]

So we have two equations to solve:

  1. \(x - 18 = 14\)
  2. \(x - 18 = -14\)

For the first equation:

\[ x - 18 = 14 \quad \Rightarrow \quad x = 14 + 18 = 32 \]

For the second equation:

\[ x - 18 = -14 \quad \Rightarrow \quad x = -14 + 18 = 4 \]

Thus, the solutions are \(x = 32\) and \(x = 4\).

The complete solution set is:

\({32, 4}\)

So the correct response is: {32, 4}.

Solve the following quadratic equation using square roots: (x+6)^2 2=8 .(1 point)
Responses

​ {−0.34,−11.66}
​ left brace negative 0.34 comma negative 11.66 right brace

{10,−22}
left brace 10 comma negative 22 right brace

{−2,−10}
left brace negative 2 comma negative 10 right brace

{−1.17,−6.83}

To solve the quadratic equation \((x + 6)^2 = 8\) using square roots, we start by taking the square root of both sides.

  1. Rewrite the equation: \[ (x + 6)^2 = 8 \]

  2. Take the square root of both sides: \[ x + 6 = \sqrt{8} \quad \text{or} \quad x + 6 = -\sqrt{8} \]

  3. Simplify \(\sqrt{8}\): \[ \sqrt{8} = 2\sqrt{2} \approx 2.83 \]

  4. Now, we have two equations to solve:

    • From \(x + 6 = 2\sqrt{2}\): \[ x = 2\sqrt{2} - 6 \approx 2.83 - 6 \approx -3.17 \]

    • From \(x + 6 = -2\sqrt{2}\): \[ x = -2\sqrt{2} - 6 \approx -2.83 - 6 \approx -8.83 \]

Putting it all together, the solutions are:

  • Approximately \(x \approx -3.17\)
  • Approximately \(x \approx -8.83\)

However, none of the choices correspond exactly to these approximations. Let's revisit the calculations:

  1. Calculate exact roots:

    • \(x = 2\sqrt{2} - 6\)
    • \(x = -2\sqrt{2} - 6\)

Using a calculator, we find:

  • \(2\sqrt{2} \approx 2.83\), thus \(x \approx -3.17\)
  • \(-2\sqrt{2} \approx -2.83\), thus \(x \approx -8.83\)

In any case, we still do not find a match with the provided choices.

None of the provided answers correspond to the calculated values. However, if we closely examine the provided answers:

  • {−0.34,−11.66}
  • {10,−22}
  • {−2,−10}
  • {−1.17,−6.83}

The closest pairs would be {−1.17,−6.83}, and are a possibility of error in values from the choices provided.

If required to select, I would go with: {−1.17,−6.83}.

If you have specific parameters or if the question has partial values, please ensure to recalculate based on end restrictions or need for square estimate limits.

Solve the following quadratic equation using square roots: (x+36)^2−38=251.(1 point)
Responses

{253,−325}
left brace 253 comma negative 325 right brace

{53,19}
left brace 53 comma 19 right brace

{17.84,−13.84}
left brace 17.84 comma negative 13.84 right brace

{−19,−53}

To solve the quadratic equation \((x + 36)^2 - 38 = 251\), we first need to simplify the equation:

  1. Add 38 to both sides:

\[ (x + 36)^2 = 251 + 38 \]

\[ (x + 36)^2 = 289 \]

  1. Now, take the square root of both sides:

\[ x + 36 = \sqrt{289} \quad \text{or} \quad x + 36 = -\sqrt{289} \]

  1. Calculate \(\sqrt{289}\):

\[ \sqrt{289} = 17 \]

  1. Now we have two equations to solve:
  • \(x + 36 = 17\)

\[ x = 17 - 36 \] \[ x = -19 \]

  • \(x + 36 = -17\)

\[ x = -17 - 36 \] \[ x = -53 \]

Thus, the solutions are \(x = -19\) and \(x = -53\).

The complete solution set is:

\({-19, -53}\)

So the correct response is: {−19,−53}.