Basic Math Examples

$h (3 z) = 2 \cdot (3 z^{2}) - 6$

Step 1

Rewrite using the commutative property of multiplication.

$3 h z = 2 \cdot (3 z^{2}) - 6$

Step 2

Multiply $3$ by $2$ .

$3 h z = 6 z^{2} - 6$

Step 3

Subtract $6 z^{2}$ from both sides of the equation.

$3 h z - 6 z^{2} = - 6$

Step 4

Add $6$ to both sides of the equation.

$3 h z - 6 z^{2} + 6 = 0$

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 $a = - 6$ , $b = 3 h$ , and $c = 6$ into the quadratic formula and solve for $z$ .

$\frac{- 3 h \pm \sqrt{{(3 h)}^{2} - 4 \cdot (- 6 \cdot 6)}}{2 \cdot - 6}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the product rule to $3 h$ .

$z = \frac{- 3 h \pm \sqrt{3^{2} h^{2} - 4 \cdot - 6 \cdot 6}}{2 \cdot - 6}$

Step 7.1.2

Raise $3$ to the power of $2$ .

$z = \frac{- 3 h \pm \sqrt{9 h^{2} - 4 \cdot - 6 \cdot 6}}{2 \cdot - 6}$

Step 7.1.3

Multiply $- 4 \cdot - 6 \cdot 6$ .

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Step 7.1.3.1

Multiply $- 4$ by $- 6$ .

$z = \frac{- 3 h \pm \sqrt{9 h^{2} + 24 \cdot 6}}{2 \cdot - 6}$

Step 7.1.3.2

Multiply $24$ by $6$ .

$z = \frac{- 3 h \pm \sqrt{9 h^{2} + 144}}{2 \cdot - 6}$

Step 7.1.4

Factor $9$ out of $9 h^{2} + 144$ .

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Step 7.1.4.1

Factor $9$ out of $9 h^{2}$ .

$z = \frac{- 3 h \pm \sqrt{9 (h^{2}) + 144}}{2 \cdot - 6}$

Step 7.1.4.2

Factor $9$ out of $144$ .

$z = \frac{- 3 h \pm \sqrt{9 h^{2} + 9 \cdot 16}}{2 \cdot - 6}$

Step 7.1.4.3

Factor $9$ out of $9 h^{2} + 9 \cdot 16$ .

$z = \frac{- 3 h \pm \sqrt{9 (h^{2} + 16)}}{2 \cdot - 6}$

Step 7.1.5

Rewrite $9 (h^{2} + 16)$ as $3^{2} (h^{2} + 4^{2})$ .

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Step 7.1.5.1

Rewrite $9$ as $3^{2}$ .

$z = \frac{- 3 h \pm \sqrt{3^{2} (h^{2} + 16)}}{2 \cdot - 6}$

Step 7.1.5.2

Rewrite $16$ as $4^{2}$ .

$z = \frac{- 3 h \pm \sqrt{3^{2} (h^{2} + 4^{2})}}{2 \cdot - 6}$

Step 7.1.6

Pull terms out from under the radical.

$z = \frac{- 3 h \pm 3 \sqrt{h^{2} + 4^{2}}}{2 \cdot - 6}$

Step 7.1.7

Raise $4$ to the power of $2$ .

$z = \frac{- 3 h \pm 3 \sqrt{h^{2} + 16}}{2 \cdot - 6}$

Step 7.2

Multiply $2$ by $- 6$ .

$z = \frac{- 3 h \pm 3 \sqrt{h^{2} + 16}}{- 12}$

Step 7.3

Simplify $\frac{- 3 h \pm 3 \sqrt{h^{2} + 16}}{- 12}$ .

$z = \frac{h \pm \sqrt{h^{2} + 16}}{4}$

Step 8

The final answer is the combination of both solutions.

$z = \frac{h + \sqrt{h^{2} + 16}}{4}$

$z = \frac{h - \sqrt{h^{2} + 16}}{4}$