Question

Solve quadratic equation by completing the square; give exact values as answers.
3z^2+5z+1
(Please show your steps)

Answer 1

Divide by 3
u get
z^2 + 5z/3 + 1/3

now

z^2 + 2(z)(5/6) + 1/3 + (5/6)^2 - (5/6)^2 =0

Answer 2

group

(z^2 + 5z/3 + (5/6)^2 ) +1/3 - (5/6)^2 = 0

Answer 3

Solving by completing the square if the coefficient isn't 1 requires dividing the coefficient of the \(x\) term not only by 2 but also the square root of the coefficient of the highest order term. After that, it is the same drill:

\[3z^2 + 5z + 1 =0\]\[3z^2+5z=-1\]
\[3z^2+5z+(\frac{5}{2\sqrt{3}})^2 = (\frac{5}{2\sqrt{3}})^2-1\]\[(\sqrt{3}z+\frac{5}{2\sqrt{3}})^2 = \frac{13}{12}\]\[(\sqrt{3}z+\frac{5}{2\sqrt{3}}) = \pm \frac{\sqrt{13}}{2\sqrt{3}}\]Clear fractions by multiplying through with \(2\sqrt{3}\)\[6z+5 = \pm \sqrt{13}\]\[z=\frac{-5\pm\sqrt{13}}{6} \approx -0.232408, -1.43426\]