Warm Up Solutions

Solve the quadratic equation by completing the square.

$6 x^{2} + 3 x + 6 = 0$

$x^{2} + \frac{1}{2} x + 1 = 0 x^{2} + \frac{1}{2} x + {(\frac{1}{2 \cdot 2})}^{2} = - 1 + {(\frac{1}{2 \cdot 2})}^{2} x^{2} + \frac{1}{2} x + \frac{1}{16} = - 1 + \frac{1}{16} {(x + \frac{1}{4})}^{2} = - \frac{15}{16} \sqrt{{(x + \frac{1}{4})}^{2}} = \pm \sqrt{- \frac{15}{16}} x + \frac{1}{4} = \pm \frac{i \sqrt{15}}{4} x = \frac{- 1 \pm i \sqrt{15}}{4}$

Note

Q: What is the first step in completing the square when a does not equal one?
A: Divide all terms by the value of a.  

Q: Classify the solution. Explain.
A: This is a complex solution because it is in the form $(a \pm b i)$ .

Solve the system of equations.

$x - z = - 2 2 x - y + z = 29 x + y + 2 z = 67$

Note

This lesson uses systems of equations to solve. See Lesson 2 for more review.