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Question

# Solve by completing the square. Check your solutions. $x^{2}-x=2$

Solution

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Step 1
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Determine the solution to the equation.

\begin{align*} x^{2} - x + \dfrac{1}{4} & = 2 + \dfrac{1}{4} && {\text {find one-half of the coefficient of x then square it; add \dfrac{1}{4} to each side}} \\ \left (x - \dfrac{1}{2} \right)^{2} & = \dfrac{9}{4} && {\text {factor the perfect square trinomial}} \\\\ \sqrt{\left (x - \dfrac{1}{2} \right)^{2}} & = \sqrt{\dfrac{9}{4}} && {\text {square root property}} \\ x - \dfrac{1}{2} & = \pm \dfrac{3}{2} \\ x - \dfrac{1}{2} + \dfrac{1}{2} & = \dfrac{1}{2} \pm \dfrac{3}{2} && {\text {add \dfrac{1}{2} to both sides of the equation}} \\ x & = \dfrac{1}{2} \pm \dfrac{3}{2} \\\\ x & = \dfrac{1}{2} + \dfrac{3}{2} && {\text{write as two equations to simplify}} \\ x & = \dfrac{4}{2} \\ x & = 2 && {\text {solution to the equation}} \\\\ x & = \dfrac{1}{2} - \dfrac{3}{2}\\ x & = -\dfrac{2}{2} \\ x & = -1 && {\text {solution to the equation}} \end{align*}

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