Basic Math Examples

$y = \frac{1}{3} \cdot ((z + 4) (z + 1))$

Step 1

Rewrite the equation as $\frac{1}{3} \cdot ((z + 4) (z + 1)) = y$ .

$\frac{1}{3} \cdot ((z + 4) (z + 1)) = y$

Step 2

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot ((z + 4) (z + 1))) = 3 y$

Step 3

Simplify the left side.

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Step 3.1

Simplify $3 (\frac{1}{3} \cdot ((z + 4) (z + 1)))$ .

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Step 3.1.1

Expand $(z + 4) (z + 1)$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$3 (\frac{1}{3} \cdot (z (z + 1) + 4 (z + 1))) = 3 y$

Step 3.1.1.2

Apply the distributive property.

$3 (\frac{1}{3} \cdot (z \cdot z + z \cdot 1 + 4 (z + 1))) = 3 y$

Step 3.1.1.3

Apply the distributive property.

$3 (\frac{1}{3} \cdot (z \cdot z + z \cdot 1 + 4 z + 4 \cdot 1)) = 3 y$

Step 3.1.2

Simplify and combine like terms.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $z$ by $z$ .

$3 (\frac{1}{3} \cdot (z^{2} + z \cdot 1 + 4 z + 4 \cdot 1)) = 3 y$

Step 3.1.2.1.2

Multiply $z$ by $1$ .

$3 (\frac{1}{3} \cdot (z^{2} + z + 4 z + 4 \cdot 1)) = 3 y$

Step 3.1.2.1.3

Multiply $4$ by $1$ .

$3 (\frac{1}{3} \cdot (z^{2} + z + 4 z + 4)) = 3 y$

Step 3.1.2.2

Add $z$ and $4 z$ .

$3 (\frac{1}{3} \cdot (z^{2} + 5 z + 4)) = 3 y$

Step 3.1.3

Apply the distributive property.

$3 (\frac{1}{3} z^{2} + \frac{1}{3} (5 z) + \frac{1}{3} \cdot 4) = 3 y$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Combine $\frac{1}{3}$ and $z^{2}$ .

$3 (\frac{z^{2}}{3} + \frac{1}{3} (5 z) + \frac{1}{3} \cdot 4) = 3 y$

Step 3.1.4.2

Multiply $\frac{1}{3} (5 z)$ .

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Step 3.1.4.2.1

Combine $5$ and $\frac{1}{3}$ .

$3 (\frac{z^{2}}{3} + \frac{5}{3} z + \frac{1}{3} \cdot 4) = 3 y$

Step 3.1.4.2.2

Combine $\frac{5}{3}$ and $z$ .

$3 (\frac{z^{2}}{3} + \frac{5 z}{3} + \frac{1}{3} \cdot 4) = 3 y$

Step 3.1.4.3

Combine $\frac{1}{3}$ and $4$ .

$3 (\frac{z^{2}}{3} + \frac{5 z}{3} + \frac{4}{3}) = 3 y$

Step 3.1.5

Apply the distributive property.

$3 \frac{z^{2}}{3} + 3 \frac{5 z}{3} + 3 (\frac{4}{3}) = 3 y$

Step 3.1.6

Simplify.

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Step 3.1.6.1

Cancel the common factor of $3$ .

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Step 3.1.6.1.1

Cancel the common factor.

$3 \frac{z^{2}}{3} + 3 \frac{5 z}{3} + 3 (\frac{4}{3}) = 3 y$

Step 3.1.6.1.2

Rewrite the expression.

$z^{2} + 3 \frac{5 z}{3} + 3 (\frac{4}{3}) = 3 y$

Step 3.1.6.2

Cancel the common factor of $3$ .

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Step 3.1.6.2.1

Cancel the common factor.

$z^{2} + 3 \frac{5 z}{3} + 3 (\frac{4}{3}) = 3 y$

Step 3.1.6.2.2

Rewrite the expression.

$z^{2} + 5 z + 3 (\frac{4}{3}) = 3 y$

Step 3.1.6.3

Cancel the common factor of $3$ .

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Step 3.1.6.3.1

Cancel the common factor.

$z^{2} + 5 z + 3 (\frac{4}{3}) = 3 y$

Step 3.1.6.3.2

Rewrite the expression.

$z^{2} + 5 z + 4 = 3 y$

Step 4

Subtract $3 y$ from both sides of the equation.

$z^{2} + 5 z + 4 - 3 y = 0$

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 $a = 1$ , $b = 5$ , and $c = 4 - 3 y$ into the quadratic formula and solve for $z$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot (4 - 3 y))}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $5$ to the power of $2$ .

$z = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot (4 - 3 y)}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4$ by $1$ .

$z = \frac{- 5 \pm \sqrt{25 - 4 \cdot (4 - 3 y)}}{2 \cdot 1}$

Step 7.1.3

Apply the distributive property.

$z = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 - 4 (- 3 y)}}{2 \cdot 1}$

Step 7.1.4

Multiply $- 4$ by $4$ .

$z = \frac{- 5 \pm \sqrt{25 - 16 - 4 (- 3 y)}}{2 \cdot 1}$

Step 7.1.5

Multiply $- 3$ by $- 4$ .

$z = \frac{- 5 \pm \sqrt{25 - 16 + 12 y}}{2 \cdot 1}$

Step 7.1.6

Subtract $16$ from $25$ .

$z = \frac{- 5 \pm \sqrt{9 + 12 y}}{2 \cdot 1}$

Step 7.1.7

Factor $3$ out of $9 + 12 y$ .

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Step 7.1.7.1

Factor $3$ out of $9$ .

$z = \frac{- 5 \pm \sqrt{3 (3) + 12 y}}{2 \cdot 1}$

Step 7.1.7.2

Factor $3$ out of $12 y$ .

$z = \frac{- 5 \pm \sqrt{3 (3) + 3 (4 y)}}{2 \cdot 1}$

Step 7.1.7.3

Factor $3$ out of $3 (3) + 3 (4 y)$ .

$z = \frac{- 5 \pm \sqrt{3 (3 + 4 y)}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$z = \frac{- 5 \pm \sqrt{3 (3 + 4 y)}}{2}$

Step 8

The final answer is the combination of both solutions.

$z = - \frac{5 - \sqrt{3 (3 + 4 y)}}{2}$

$z = - \frac{5 + \sqrt{3 (4 y + 3)}}{2}$