Solve 6z^2=z+222

$$6 z ^ { 2 } = z + 222$$

Solve for z

$$z=-6$$

$$z = \frac{37}{6} = 6\frac{1}{6} \approx 6.166666667$$

Steps Using Factoring By Grouping

Subtract z from both sides.

$$6z^{2}-z=222$$

Subtract 222 from both sides.

$$6z^{2}-z-222=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6z^{2}+az+bz-222. To find a and b, set up a system to be solved.

$$a+b=-1 ab=6\left(-222\right)=-1332$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1332.

$$1,-1332 2,-666 3,-444 4,-333 6,-222 9,-148 12,-111 18,-74 36,-37$$

Calculate the sum for each pair.

$$1-1332=-1331 2-666=-664 3-444=-441 4-333=-329 6-222=-216 9-148=-139 12-111=-99 18-74=-56 36-37=-1$$

The solution is the pair that gives sum -1.

$$a=-37 b=36$$

Rewrite 6z^{2}-z-222 as \left(6z^{2}-37z\right)+\left(36z-222\right).

$$\left(6z^{2}-37z\right)+\left(36z-222\right)$$

Factor out z in the first and 6 in the second group.

$$z\left(6z-37\right)+6\left(6z-37\right)$$

Factor out common term 6z-37 by using distributive property.

$$\left(6z-37\right)\left(z+6\right)$$

To find equation solutions, solve 6z-37=0 and z+6=0.

$$z=\frac{37}{6} z=-6$$

Steps Using the Quadratic Formula

Subtract z from both sides.

$$6z^{2}-z=222$$

Subtract 222 from both sides.

$$6z^{2}-z-222=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -1 for b, and -222 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-1\right)±\sqrt{1-4\times 6\left(-222\right)}}{2\times 6}$$

Multiply -4 times 6.

$$z=\frac{-\left(-1\right)±\sqrt{1-24\left(-222\right)}}{2\times 6}$$

Multiply -24 times -222.

$$z=\frac{-\left(-1\right)±\sqrt{1+5328}}{2\times 6}$$

Add 1 to 5328.

$$z=\frac{-\left(-1\right)±\sqrt{5329}}{2\times 6}$$

Take the square root of 5329.

$$z=\frac{-\left(-1\right)±73}{2\times 6}$$

The opposite of -1 is 1.

$$z=\frac{1±73}{2\times 6}$$

Multiply 2 times 6.

$$z=\frac{1±73}{12}$$

Now solve the equation z=\frac{1±73}{12} when ± is plus. Add 1 to 73.

$$z=\frac{74}{12}$$

Reduce the fraction \frac{74}{12} to lowest terms by extracting and canceling out 2.

$$z=\frac{37}{6}$$

Now solve the equation z=\frac{1±73}{12} when ± is minus. Subtract 73 from 1.

$$z=-\frac{72}{12}$$

Divide -72 by 12.

$$z=-6$$

The equation is now solved.

$$z=\frac{37}{6} z=-6$$

Steps for Completing the Square

Subtract z from both sides.

$$6z^{2}-z=222$$

Divide both sides by 6.

$$\frac{6z^{2}-z}{6}=\frac{222}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$z^{2}-\frac{1}{6}z=\frac{222}{6}$$

Divide 222 by 6.

$$z^{2}-\frac{1}{6}z=37$$

Divide -\frac{1}{6}, the coefficient of the x term, by 2 to get -\frac{1}{12}. Then add the square of -\frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{1}{6}z+\left(-\frac{1}{12}\right)^{2}=37+\left(-\frac{1}{12}\right)^{2}$$

Square -\frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{1}{6}z+\frac{1}{144}=37+\frac{1}{144}$$

Add 37 to \frac{1}{144}.

$$z^{2}-\frac{1}{6}z+\frac{1}{144}=\frac{5329}{144}$$

Factor z^{2}-\frac{1}{6}z+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{1}{12}\right)^{2}=\frac{5329}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{1}{12}\right)^{2}}=\sqrt{\frac{5329}{144}}$$

Simplify.

$$z-\frac{1}{12}=\frac{73}{12} z-\frac{1}{12}=-\frac{73}{12}$$

Add \frac{1}{12} to both sides of the equation.

$$z=\frac{37}{6} z=-6$$