Precalculus Examples

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

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 2.1.1

Multiply by $1$ .

$3 z^{2} + 1 z - 4 = 0$

Step 2.1.2

Rewrite $1$ as $- 3$ plus $4$

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$3 z (z - 1) + 4 (z - 1) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $z - 1$ .

$(z - 1) (3 z + 4) = 0$

Step 3

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

$z - 1 = 0$

$3 z + 4 = 0$

Step 4

Set $z - 1$ equal to $0$ and solve for $z$ .

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Step 4.1

Set $z - 1$ equal to $0$ .

$z - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$z = 1$

Step 5

Set $3 z + 4$ equal to $0$ and solve for $z$ .

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Step 5.1

Set $3 z + 4$ equal to $0$ .

$3 z + 4 = 0$

Step 5.2

Solve $3 z + 4 = 0$ for $z$ .

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Step 5.2.1

Subtract $4$ from both sides of the equation.

$3 z = - 4$

Step 5.2.2

Divide each term in $3 z = - 4$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 z = - 4$ by $3$ .

$\frac{3 z}{3} = \frac{- 4}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{- 4}{3}$

Step 5.2.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 4}{3}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$z = - \frac{4}{3}$

Step 6

The final solution is all the values that make $(z - 1) (3 z + 4) = 0$ true.

$z = 1, - \frac{4}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = 1, - \frac{4}{3}$

Decimal Form:

$z = 1, - 1.\overline{3}$

Mixed Number Form:

$z = 1, - 1 \frac{1}{3}$