Basic Math Examples

$30 z = 9 z^{2} + 25$

Step 1

Subtract $9 z^{2}$ from both sides of the equation.

$30 z - 9 z^{2} = 25$

Step 2

Subtract $25$ from both sides of the equation.

$30 z - 9 z^{2} - 25 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $30 z - 9 z^{2} - 25$ .

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Step 3.1.1

Reorder $30 z$ and $- 9 z^{2}$ .

$- 9 z^{2} + 30 z - 25 = 0$

Step 3.1.2

Factor $- 1$ out of $- 9 z^{2}$ .

$- (9 z^{2}) + 30 z - 25 = 0$

Step 3.1.3

Factor $- 1$ out of $30 z$ .

$- (9 z^{2}) - (- 30 z) - 25 = 0$

Step 3.1.4

Rewrite $- 25$ as $- 1 (25)$ .

$- (9 z^{2}) - (- 30 z) - 1 \cdot 25 = 0$

Step 3.1.5

Factor $- 1$ out of $- (9 z^{2}) - (- 30 z)$ .

$- (9 z^{2} - 30 z) - 1 \cdot 25 = 0$

Step 3.1.6

Factor $- 1$ out of $- (9 z^{2} - 30 z) - 1 (25)$ .

$- (9 z^{2} - 30 z + 25) = 0$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rewrite $9 z^{2}$ as ${(3 z)}^{2}$ .

$- ({(3 z)}^{2} - 30 z + 25) = 0$

Step 3.2.2

Rewrite $25$ as $5^{2}$ .

$- ({(3 z)}^{2} - 30 z + 5^{2}) = 0$

Step 3.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$30 z = 2 \cdot (3 z) \cdot 5$

Step 3.2.4

Rewrite the polynomial.

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

Step 3.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 z$ and $b = 5$ .

$- {(3 z - 5)}^{2} = 0$

Step 4

Divide each term in $- {(3 z - 5)}^{2} = 0$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- {(3 z - 5)}^{2} = 0$ by $- 1$ .

$\frac{- {(3 z - 5)}^{2}}{- 1} = \frac{0}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{{(3 z - 5)}^{2}}{1} = \frac{0}{- 1}$

Step 4.2.2

Divide ${(3 z - 5)}^{2}$ by $1$ .

${(3 z - 5)}^{2} = \frac{0}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 1$ .

${(3 z - 5)}^{2} = 0$

Step 5

Set the $3 z - 5$ equal to $0$ .

$3 z - 5 = 0$

Step 6

Solve for $z$ .

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Step 6.1

Add $5$ to both sides of the equation.

$3 z = 5$

Step 6.2

Divide each term in $3 z = 5$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 z = 5$ by $3$ .

$\frac{3 z}{3} = \frac{5}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{5}{3}$

Step 6.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = \frac{5}{3}$

Decimal Form:

$z = 1.\overline{6}$

Mixed Number Form:

$z = 1 \frac{2}{3}$