Question

Solve the quadratic equation by completing the square. 2+x=6x^2Completing the square gives us: (x- Answer )^2 = AnswerEnter your solutions below from smallest to largest. If a solution is repeated type that answer for both values of x. If your answer is not an integer then type it as a decimal rounded to the nearest hundredth.x=Answer and x=Answer

Answer 1

Subtracting x from the given equation we get:

`6x^2-x=2+x-x=2.`

Dividing by 6 we get:

`x^2-(1)/(6)x=(1)/(3)\text{.}`

Notice that:

`x^2-(1)/(6)x=x^2+2(1)(-(1)/(12))x\text{.}`

Therefore:

`x^2+2(1)(-(1)/(12))x=(1)/(3)`

Adding (-1/12)² from the above equation we get:

`\begin{gathered} x^2+2(1)(-(1)/(12))x+(-(1)/(12))^2=(1)/(3)+(-(1)/(12))^2, \\ (x-(1)/(12))^2=(1)/(3)+(1)/(144), \\ (x-(1)/(12))^2=(49)/(144). \end{gathered}`

Solving for x we get:

`\begin{gathered} (x-(1)/(12))^2=(7^2)/(12^2), \\ x-(1)/(12)^{}=\pm\frac{7^{}}{12^{}}, \\ x=(1)/(12)\pm\frac{7^{}}{12^{}}, \\ x=(2)/(3)\text{ or x=-}\frac{1\text{ }}{2}\text{.} \end{gathered}`

Answer: Completing the square gives us:

`(x-(1)/(12))^2=(49)/(144).`