Basic Math Examples

$\frac{4 z + 1}{3} = \frac{3 \cdot - 1}{z}$

Step 1

Simplify both sides.

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Step 1.1

Multiply $3$ by $- 1$ .

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

Step 1.2

Move the negative in front of the fraction.

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

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(4 z + 1) z = 3 \cdot - 3$

Step 3

Solve the equation for $z$ .

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Step 3.1

Simplify $(4 z + 1) z$ .

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Step 3.1.1

Apply the distributive property.

$4 z \cdot z + 1 z = 3 \cdot - 3$

Step 3.1.2

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

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Step 3.1.2.1

Move $z$ .

$4 (z \cdot z) + 1 z = 3 \cdot - 3$

Step 3.1.2.2

Multiply $z$ by $z$ .

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

Step 3.1.3

Multiply $z$ by $1$ .

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

Step 3.2

Multiply $3$ by $- 3$ .

$4 z^{2} + z = - 9$

Step 3.3

Add $9$ to both sides of the equation.

$4 z^{2} + z + 9 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5

Substitute the values $a = 4$ , $b = 1$ , and $c = 9$ into the quadratic formula and solve for $z$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (4 \cdot 9)}}{2 \cdot 4}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

One to any power is one.

$z = \frac{- 1 \pm \sqrt{1 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Step 3.6.1.2

Multiply $- 4 \cdot 4 \cdot 9$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{- 1 \pm \sqrt{1 - 16 \cdot 9}}{2 \cdot 4}$

Step 3.6.1.2.2

Multiply $- 16$ by $9$ .

$z = \frac{- 1 \pm \sqrt{1 - 144}}{2 \cdot 4}$

Step 3.6.1.3

Subtract $144$ from $1$ .

$z = \frac{- 1 \pm \sqrt{- 143}}{2 \cdot 4}$

Step 3.6.1.4

Rewrite $- 143$ as $- 1 (143)$ .

$z = \frac{- 1 \pm \sqrt{- 1 \cdot 143}}{2 \cdot 4}$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (143)}$ as $\sqrt{- 1} \cdot \sqrt{143}$ .

$z = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{143}}{2 \cdot 4}$

Step 3.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$z = \frac{- 1 \pm i \sqrt{143}}{2 \cdot 4}$

Step 3.6.2

Multiply $2$ by $4$ .

$z = \frac{- 1 \pm i \sqrt{143}}{8}$

Step 3.7

The final answer is the combination of both solutions.

$z = - \frac{1 - i \sqrt{143}}{8}, - \frac{1 + i \sqrt{143}}{8}$