Basic Math Examples

$16 = 3 (z - 1) (z - 7)$

Step 1

Rewrite the equation as $3 (z - 1) (z - 7) = 16$ .

$3 (z - 1) (z - 7) = 16$

Step 2

Divide each term in $3 (z - 1) (z - 7) = 16$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 (z - 1) (z - 7) = 16$ by $3$ .

$\frac{3 (z - 1) (z - 7)}{3} = \frac{16}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 (z - 1) (z - 7)}{3} = \frac{16}{3}$

Step 2.2.1.2

Divide $(z - 1) (z - 7)$ by $1$ .

$(z - 1) (z - 7) = \frac{16}{3}$

Step 2.2.2

Expand $(z - 1) (z - 7)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$z (z - 7) - 1 (z - 7) = \frac{16}{3}$

Step 2.2.2.2

Apply the distributive property.

$z \cdot z + z \cdot - 7 - 1 (z - 7) = \frac{16}{3}$

Step 2.2.2.3

Apply the distributive property.

$z \cdot z + z \cdot - 7 - 1 z - 1 \cdot - 7 = \frac{16}{3}$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $z$ by $z$ .

$z^{2} + z \cdot - 7 - 1 z - 1 \cdot - 7 = \frac{16}{3}$

Step 2.2.3.1.2

Move $- 7$ to the left of $z$ .

$z^{2} - 7 \cdot z - 1 z - 1 \cdot - 7 = \frac{16}{3}$

Step 2.2.3.1.3

Rewrite $- 1 z$ as $- z$ .

$z^{2} - 7 z - z - 1 \cdot - 7 = \frac{16}{3}$

Step 2.2.3.1.4

Multiply $- 1$ by $- 7$ .

$z^{2} - 7 z - z + 7 = \frac{16}{3}$

Step 2.2.3.2

Subtract $z$ from $- 7 z$ .

$z^{2} - 8 z + 7 = \frac{16}{3}$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Subtract $\frac{16}{3}$ from both sides of the equation.

$z^{2} - 8 z + 7 - \frac{16}{3} = 0$

Step 3.2

Simplify $z^{2} - 8 z + 7 - \frac{16}{3}$ .

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Step 3.2.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$z^{2} - 8 z + 7 \cdot \frac{3}{3} - \frac{16}{3} = 0$

Step 3.2.2

Combine $7$ and $\frac{3}{3}$ .

$z^{2} - 8 z + \frac{7 \cdot 3}{3} - \frac{16}{3} = 0$

Step 3.2.3

Combine the numerators over the common denominator.

$z^{2} - 8 z + \frac{7 \cdot 3 - 16}{3} = 0$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $7$ by $3$ .

$z^{2} - 8 z + \frac{21 - 16}{3} = 0$

Step 3.2.4.2

Subtract $16$ from $21$ .

$z^{2} - 8 z + \frac{5}{3} = 0$

Step 4

Multiply through by the least common denominator $3$ , then simplify.

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Step 4.1

Apply the distributive property.

$3 z^{2} + 3 (- 8 z) + 3 (\frac{5}{3}) = 0$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $- 8$ by $3$ .

$3 z^{2} - 24 z + 3 (\frac{5}{3}) = 0$

Step 4.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.1

Cancel the common factor.

$3 z^{2} - 24 z + 3 (\frac{5}{3}) = 0$

Step 4.2.2.2

Rewrite the expression.

$3 z^{2} - 24 z + 5 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 3$ , $b = - 24$ , and $c = 5$ into the quadratic formula and solve for $z$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (3 \cdot 5)}}{2 \cdot 3}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 24$ to the power of $2$ .

$z = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$z = \frac{24 \pm \sqrt{576 - 12 \cdot 5}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $5$ .

$z = \frac{24 \pm \sqrt{576 - 60}}{2 \cdot 3}$

Step 7.1.3

Subtract $60$ from $576$ .

$z = \frac{24 \pm \sqrt{516}}{2 \cdot 3}$

Step 7.1.4

Rewrite $516$ as $2^{2} \cdot 129$ .

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Step 7.1.4.1

Factor $4$ out of $516$ .

$z = \frac{24 \pm \sqrt{4 (129)}}{2 \cdot 3}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$z = \frac{24 \pm \sqrt{2^{2} \cdot 129}}{2 \cdot 3}$

Step 7.1.5

Pull terms out from under the radical.

$z = \frac{24 \pm 2 \sqrt{129}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$z = \frac{24 \pm 2 \sqrt{129}}{6}$

Step 7.3

Simplify $\frac{24 \pm 2 \sqrt{129}}{6}$ .

$z = \frac{12 \pm \sqrt{129}}{3}$

Step 8

The final answer is the combination of both solutions.

$z = \frac{12 + \sqrt{129}}{3}, \frac{12 - \sqrt{129}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$z = \frac{12 + \sqrt{129}}{3}, \frac{12 - \sqrt{129}}{3}$

Decimal Form:

$z = 7.78593889 \dots, 0.21406110 \dots$