Algebra Examples

$\frac{17 z^{3} + 17 z^{2}}{34 z^{3} - 51 z^{2}}$

Step 1

Set the denominator in $\frac{17 z^{3} + 17 z^{2}}{34 z^{3} - 51 z^{2}}$ equal to $0$ to find where the expression is undefined.

$34 z^{3} - 51 z^{2} = 0$

Step 2

Solve for $z$ .

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Step 2.1

Factor $17 z^{2}$ out of $34 z^{3} - 51 z^{2}$ .

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Step 2.1.1

Factor $17 z^{2}$ out of $34 z^{3}$ .

$17 z^{2} (2 z) - 51 z^{2} = 0$

Step 2.1.2

Factor $17 z^{2}$ out of $- 51 z^{2}$ .

$17 z^{2} (2 z) + 17 z^{2} (- 3) = 0$

Step 2.1.3

Factor $17 z^{2}$ out of $17 z^{2} (2 z) + 17 z^{2} (- 3)$ .

$17 z^{2} (2 z - 3) = 0$

Step 2.2

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

$z^{2} = 0$

$2 z - 3 = 0$

Step 2.3

Set $z^{2}$ equal to $0$ and solve for $z$ .

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Step 2.3.1

Set $z^{2}$ equal to $0$ .

$z^{2} = 0$

Step 2.3.2

Solve $z^{2} = 0$ for $z$ .

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Step 2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \pm \sqrt{0}$

Step 2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$z = \pm \sqrt{0^{2}}$

Step 2.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$z = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$z = 0$

Step 2.4

Set $2 z - 3$ equal to $0$ and solve for $z$ .

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Step 2.4.1

Set $2 z - 3$ equal to $0$ .

$2 z - 3 = 0$

Step 2.4.2

Solve $2 z - 3 = 0$ for $z$ .

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Step 2.4.2.1

Add $3$ to both sides of the equation.

$2 z = 3$

Step 2.4.2.2

Divide each term in $2 z = 3$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 z = 3$ by $2$ .

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

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

Divide $z$ by $1$ .

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

Step 2.5

The final solution is all the values that make $17 z^{2} (2 z - 3) = 0$ true.

$z = 0, \frac{3}{2}$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$z = 0, z = \frac{3}{2}$

Step 4