Solve by Factoring (z^2)/6-z/2-84=0

$\frac{z^{2}}{6} - \frac{z}{2} - 84 = 0$

Step 1

To write $- \frac{z}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{z^{2}}{6} - \frac{z}{2} \cdot \frac{3}{3} - 84 = 0$

Step 2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{z}{2}$ by $\frac{3}{3}$ .

$\frac{z^{2}}{6} - \frac{z \cdot 3}{2 \cdot 3} - 84 = 0$

Multiply $2$ by $3$ .

$\frac{z^{2}}{6} - \frac{z \cdot 3}{6} - 84 = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{z^{2} - z \cdot 3}{6} - 84 = 0$

Step 4

Simplify the numerator.

Factor $z$ out of $z^{2} - z \cdot 3$ .

Factor $z$ out of $z^{2}$ .

$\frac{z \cdot z - z \cdot 3}{6} - 84 = 0$

Factor $z$ out of $- z \cdot 3$ .

$\frac{z \cdot z + z (- 1 \cdot 3)}{6} - 84 = 0$

Factor $z$ out of $z \cdot z + z (- 1 \cdot 3)$ .

$\frac{z (z - 1 \cdot 3)}{6} - 84 = 0$

Multiply $- 1$ by $3$ .

$\frac{z (z - 3)}{6} - 84 = 0$

Step 5

To write $- 84$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{z (z - 3)}{6} - 84 \cdot \frac{6}{6} = 0$

Step 6

Combine $- 84$ and $\frac{6}{6}$ .

$\frac{z (z - 3)}{6} + \frac{- 84 \cdot 6}{6} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{z (z - 3) - 84 \cdot 6}{6} = 0$

Step 8

Simplify the numerator.

Apply the distributive property.

$\frac{z \cdot z + z \cdot - 3 - 84 \cdot 6}{6} = 0$

Multiply $z$ by $z$ .

$\frac{z^{2} + z \cdot - 3 - 84 \cdot 6}{6} = 0$

Move $- 3$ to the left of $z$ .

$\frac{z^{2} - 3 z - 84 \cdot 6}{6} = 0$

Multiply $- 84$ by $6$ .

$\frac{z^{2} - 3 z - 504}{6} = 0$

Factor $z^{2} - 3 z - 504$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 504$ and whose sum is $- 3$ .

$- 24, 21$

Write the factored form using these integers.

$\frac{(z - 24) (z + 21)}{6} = 0$

Step 9

Set the numerator equal to zero.

$(z - 24) (z + 21) = 0$

Step 10

Solve the equation for $z$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z - 24 = 0$

$z + 21 = 0$

Set $z - 24$ equal to $0$ and solve for $z$ .

Set $z - 24$ equal to $0$ .

$z - 24 = 0$

Add $24$ to both sides of the equation.

$z = 24$

Set $z + 21$ equal to $0$ and solve for $z$ .

Set $z + 21$ equal to $0$ .

$z + 21 = 0$

Subtract $21$ from both sides of the equation.

$z = - 21$

The final solution is all the values that make $(z - 24) (z + 21) = 0$ true.

$z = 24, - 21$

Solve by Factoring (z^2)/6-z/2-84=0

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