Question

Solve the equation by completing square method.

$w^2=5w-1$

Short answer

The solutions are$w=\frac{5}{2}\pm \frac{\sqrt{21}}{2}$.

Step-by-step solution

Given Information    

The Given equation is $w^2=5w-1$.

Isolate the variable terms on one side and the constant terms on the other. 

$w^2-5w=-1$

Here, $b=-5$.

So,

$$\begin{gathered} ={\left(\frac{1}{2}b\right)}^2 \\ ={\left(\frac{1}{2}(-5)\right)}^2 \\ ={\left(\frac{-5}{2}\right)}^2 \\ =\frac{25}{4} \end{gathered}$$

Add $\frac{25}{4}$both sides of equation.

$$\begin{gathered} w^2-5w+\frac{25}{4}=-1+\frac{25}{4} \\ {w}^2-5w+\frac{25}{4}=\frac{25-4}{4} \\ {w}^2-5w+\frac{25}{4}=\frac{21}{4} \end{gathered}$$

Factor the perfect square trinomial as a binomial square.     

${\left(w-\frac{5}{2}\right)}^2=\frac{21}{4}$

Use the square root property.

$$\begin{gathered} w-\frac{5}{2}=\pm \sqrt{\frac{21}{4}} \\ w-\frac{5}{2}=\pm \frac{\sqrt{21}}{2} \end{gathered}$$

Solve for w.

$w=\frac{5}{2}\pm \frac{\sqrt{21}}{2}$

Rewrite to show two solutions.

$$\begin{gathered} w=\frac{5}{2}+\frac{\sqrt{21}}{2}, \\ w=\frac{5}{2}-\frac{\sqrt{21}}{2} \end{gathered}$$