Question

Solve 22 – x = 4 by completing the square.(Use a comma to separate multiple solutions.)

Answer 1

To complete the square, remember the formula for a square binomial:

`(x+a)^2=x^2+2ax+a^2`

The coefficient of the linear term is 2a. In this case:

`x^2-x=4`

We can see that the coefficient of the linear term on the left member of the equation is -1. Then:

`\begin{gathered} 2a=-1 \\ \Rightarrow a=-(1)/(2) \\ \Rightarrow a^2=(1)/(4) \end{gathered}`

Add and substract 1/4 to the left member of the equation:

`x^2-x+(1)/(4)-(1)/(4)=4`

Since the first three terms of the left member correspond to a perfect square trinomial, then we can rewrite it as a square binomial:

`\begin{gathered} x^2-x+(1)/(4)=(x-(1)/(2))^2 \\ \Rightarrow(x-(1)/(2))^2-(1)/(4)=4 \end{gathered}`

Add 1/4 to both sides of the equation:

`\begin{gathered} \Rightarrow(x-(1)/(2))^2=4+(1)/(4) \\ \Rightarrow(x-(1)/(2))^2=(17)/(4) \end{gathered}`

Take the square root to both sides of the equation:

`\begin{gathered} \Rightarrow\sqrt[]{(x-(1)/(2))^2}=\sqrt[]{(17)/(4)} \\ \Rightarrow x-(1)/(2)=\pm\frac{\sqrt[]{17}}{2} \\ \Rightarrow x=\pm\frac{\sqrt[]{17}}{2}+(1)/(2) \end{gathered}`

Therefore, the answer is:

`x=\pm\frac{\sqrt[]{17}}{2}+(1)/(2)`