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

Question

𝑥
2
−
22
𝑥
+
1
=
15
.(1 point)
Responses

x=11+14−−√, 11−14−−√
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
=
−
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
=
−
11
+
3
15
,

−
11
−
3
15
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−−√

Answer 1

To solve the equation $x^2-22x+1=15$ by completing the square, follow these steps:

1. Rearrange the equation: $$x^2-22x+1-15=0$$ Simplifying gives us: $$x^2-22x-14=0$$

2. Move the constant term to the other side: $$x^2-22x=14$$

3. Complete the square:

   • Take the coefficient of $x$ (which is $-22$), divide it by 2, and square it: $${(-\frac{22}{2})}^2=(-11{)}^2=121$$

   • Add and subtract $121$ on the left-hand side: $$x^2-22x+121-121=14$$ This simplifies to: $$(x-11{)}^2-121=14$$

4. Rewrite the equation: $$(x-11{)}^2=14+121$$ Which simplifies to: $$(x-11{)}^2=135$$

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

6. Solve for $x$: $$x=11\pm \sqrt{135}$$

Now we can rewrite $\sqrt{135}$ as $\sqrt{9\times 15}=3\sqrt{15}$, giving us:

$$x=11\pm 3\sqrt{15}$$

Thus, the solutions are: $$x=11+3\sqrt{15},x=11-3\sqrt{15}$$

So the correct response is: $$x=11+3\sqrt{15},11-3\sqrt{15}$$