Pre-Algebra Examples

$3 z^{2} + 7 z - 6 = (4 z + 1) (z - 6)$

Step 1

Simplify $(4 z + 1) (z - 6)$ .

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Step 1.1

Rewrite.

$3 z^{2} + 7 z - 6 = 0 + 0 + (4 z + 1) (z - 6)$

Step 1.2

Simplify by adding zeros.

$3 z^{2} + 7 z - 6 = (4 z + 1) (z - 6)$

Step 1.3

Expand $(4 z + 1) (z - 6)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 z^{2} + 7 z - 6 = 4 z (z - 6) + 1 (z - 6)$

Step 1.3.2

Apply the distributive property.

$3 z^{2} + 7 z - 6 = 4 z \cdot z + 4 z \cdot - 6 + 1 (z - 6)$

Step 1.3.3

Apply the distributive property.

$3 z^{2} + 7 z - 6 = 4 z \cdot z + 4 z \cdot - 6 + 1 z + 1 \cdot - 6$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $z$ by $z$ by adding the exponents.

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Step 1.4.1.1.1

Move $z$ .

$3 z^{2} + 7 z - 6 = 4 (z \cdot z) + 4 z \cdot - 6 + 1 z + 1 \cdot - 6$

Step 1.4.1.1.2

Multiply $z$ by $z$ .

$3 z^{2} + 7 z - 6 = 4 z^{2} + 4 z \cdot - 6 + 1 z + 1 \cdot - 6$

Step 1.4.1.2

Multiply $- 6$ by $4$ .

$3 z^{2} + 7 z - 6 = 4 z^{2} - 24 z + 1 z + 1 \cdot - 6$

Step 1.4.1.3

Multiply $z$ by $1$ .

$3 z^{2} + 7 z - 6 = 4 z^{2} - 24 z + z + 1 \cdot - 6$

Step 1.4.1.4

Multiply $- 6$ by $1$ .

$3 z^{2} + 7 z - 6 = 4 z^{2} - 24 z + z - 6$

Step 1.4.2

Add $- 24 z$ and $z$ .

$3 z^{2} + 7 z - 6 = 4 z^{2} - 23 z - 6$

Step 2

Move all terms containing $z$ to the left side of the equation.

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Step 2.1

Subtract $4 z^{2}$ from both sides of the equation.

$3 z^{2} + 7 z - 6 - 4 z^{2} = - 23 z - 6$

Step 2.2

Add $23 z$ to both sides of the equation.

$3 z^{2} + 7 z - 6 - 4 z^{2} + 23 z = - 6$

Step 2.3

Subtract $4 z^{2}$ from $3 z^{2}$ .

$- z^{2} + 7 z - 6 + 23 z = - 6$

Step 2.4

Add $7 z$ and $23 z$ .

$- z^{2} + 30 z - 6 = - 6$

Step 3

Add $6$ to both sides of the equation.

$- z^{2} + 30 z - 6 + 6 = 0$

Step 4

Combine the opposite terms in $- z^{2} + 30 z - 6 + 6$ .

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Step 4.1

Add $- 6$ and $6$ .

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

Step 4.2

Add $- z^{2} + 30 z$ and $0$ .

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

Step 5

Factor $- z$ out of $- z^{2} + 30 z$ .

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Step 5.1

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

$- z \cdot z + 30 z = 0$

Step 5.2

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

$- z \cdot z - z \cdot - 30 = 0$

Step 5.3

Factor $- z$ out of $- z (z) - z (- 30)$ .

$- z (z - 30) = 0$

Step 6

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

$z = 0$

$z - 30 = 0$

Step 7

Set $z$ equal to $0$ .

$z = 0$

Step 8

Set $z - 30$ equal to $0$ and solve for $z$ .

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Step 8.1

Set $z - 30$ equal to $0$ .

$z - 30 = 0$

Step 8.2

Add $30$ to both sides of the equation.

$z = 30$

Step 9

The final solution is all the values that make $- z (z - 30) = 0$ true.

$z = 0, 30$