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Complete the square to solve the following quadratic equation: x2−22x+1=15.(1 point)

Responses

x=11+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 End Root

x=−11+14−−√, −11−14−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

x=−11+315−−√, −11−315−−√
x equals negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

x=11+315−−√, 11−315−−√

1 answer

To solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we first rearrange the equation:

\[ x^2 - 22x + 1 - 15 = 0 \]

This simplifies to:

\[ x^2 - 22x - 14 = 0 \]

Next, we need to complete the square for the quadratic expression \(x^2 - 22x\).

  1. Take half of the coefficient of \(x\), which is \(-22\), and square it:

\[ \left(\frac{-22}{2}\right)^2 = (-11)^2 = 121 \]

  1. Add and subtract this square inside the equation:

\[ x^2 - 22x + 121 - 121 - 14 = 0 \]

This can be rewritten as:

\[ (x - 11)^2 - 135 = 0 \]

  1. Now, isolate the perfect square:

\[ (x - 11)^2 = 135 \]

  1. Take the square root of both sides:

\[ x - 11 = \pm \sqrt{135} \]

  1. Finally, add 11 to both sides:

\[ x = 11 \pm \sqrt{135} \]

Thus, the solutions are:

\[ x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135} \]

Now we can simplify \(\sqrt{135}\):

\[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]

So, the final answers are:

\[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]

Therefore, the correct response is:

x = 11 + 3√15, 11 - 3√15.