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2z^{2}-z+6=4\left(z+1\right)
Use the distributive property to multiply z by 2z-1.
2z^{2}-z+6=4z+4
Use the distributive property to multiply 4 by z+1.
2z^{2}-z+6-4z=4
Subtract 4z from both sides.
2z^{2}-5z+6=4
Combine -z and -4z to get -5z.
2z^{2}-5z+6-4=0
Subtract 4 from both sides.
2z^{2}-5z+2=0
Subtract 4 from 6 to get 2.
z=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\times 2}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
z=\frac{-\left(-5\right)±\sqrt{25-4\times 2\times 2}}{2\times 2}
Square -5.
z=\frac{-\left(-5\right)±\sqrt{25-8\times 2}}{2\times 2}
Multiply -4 times 2.
z=\frac{-\left(-5\right)±\sqrt{25-16}}{2\times 2}
Multiply -8 times 2.
z=\frac{-\left(-5\right)±\sqrt{9}}{2\times 2}
Add 25 to -16.
z=\frac{-\left(-5\right)±3}{2\times 2}
Take the square root of 9.
z=\frac{5±3}{2\times 2}
The opposite of -5 is 5.
z=\frac{5±3}{4}
Multiply 2 times 2.
z=\frac{8}{4}
Now solve the equation z=\frac{5±3}{4} when ± is plus. Add 5 to 3.
z=2
Divide 8 by 4.
z=\frac{2}{4}
Now solve the equation z=\frac{5±3}{4} when ± is minus. Subtract 3 from 5.
z=\frac{1}{2}
Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.
z=2 z=\frac{1}{2}
The equation is now solved.
2z^{2}-z+6=4\left(z+1\right)
Use the distributive property to multiply z by 2z-1.
2z^{2}-z+6=4z+4
Use the distributive property to multiply 4 by z+1.
2z^{2}-z+6-4z=4
Subtract 4z from both sides.
2z^{2}-5z+6=4
Combine -z and -4z to get -5z.
2z^{2}-5z=4-6
Subtract 6 from both sides.
2z^{2}-5z=-2
Subtract 6 from 4 to get -2.
\frac{2z^{2}-5z}{2}=-\frac{2}{2}
Divide both sides by 2.
z^{2}-\frac{5}{2}z=-\frac{2}{2}
Dividing by 2 undoes the multiplication by 2.
z^{2}-\frac{5}{2}z=-1
Divide -2 by 2.
z^{2}-\frac{5}{2}z+\left(-\frac{5}{4}\right)^{2}=-1+\left(-\frac{5}{4}\right)^{2}
Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
z^{2}-\frac{5}{2}z+\frac{25}{16}=-1+\frac{25}{16}
Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.
z^{2}-\frac{5}{2}z+\frac{25}{16}=\frac{9}{16}
Add -1 to \frac{25}{16}.
\left(z-\frac{5}{4}\right)^{2}=\frac{9}{16}
Factor z^{2}-\frac{5}{2}z+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(z-\frac{5}{4}\right)^{2}}=\sqrt{\frac{9}{16}}
Take the square root of both sides of the equation.
z-\frac{5}{4}=\frac{3}{4} z-\frac{5}{4}=-\frac{3}{4}
Simplify.
z=2 z=\frac{1}{2}
Add \frac{5}{4} to both sides of the equation.