Completing the Square & the Quadratic Formula Quick Check

Question

2 of 52 of 5 Items
Question
Complete the square to solve the following quadratic equation: x2−22x+1=15.(1 point)
Responses

x=11+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 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+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
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Answer 1

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, let's follow these steps:

1. Move the constant to the other side of the equation: \[ x^2 - 22x + 1 - 15 = 0 \implies x^2 - 22x - 14 = 0 \]

2. Rearrange and isolate the \( x^2 \) and \( x \) terms: \[ x^2 - 22x = 14 \]

3. Complete the square:

   • Take half of the coefficient of \( x \) (which is -22), square it, and add to both sides. Half of -22 is -11 and \((-11)^2 = 121\). \[ x^2 - 22x + 121 = 14 + 121 \] \[ x^2 - 22x + 121 = 135 \]

4. Rewrite the left side as a squared term: \[ (x - 11)^2 = 135 \]

5. Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]

6. Solve for \( x \): \[ x = 11 \pm \sqrt{135} \]

So the solutions are: \[ x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135} \]

Now, simplifying \(\sqrt{135}\): \[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]

Thus, we substitute back: \[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]

Therefore, the correct answer is: \[ x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \]

Hence, the correct response from the options would be:

• x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}.