Solve by Completing the Square 3z^2+5z-2=0

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

Step 1

Add $2$ to both sides of the equation.

$3 z^{2} + 5 z = 2$

Step 2

Divide each term in $3 z^{2} + 5 z = 2$ by $3$ and simplify.

Divide each term in $3 z^{2} + 5 z = 2$ by $3$ .

$\frac{3 z^{2}}{3} + \frac{5 z}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z^{2}}{3} + \frac{5 z}{3} = \frac{2}{3}$

Divide $z^{2}$ by $1$ .

$z^{2} + \frac{5 z}{3} = \frac{2}{3}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{5}{6})}^{2}$

Step 4

Add the term to each side of the equation.

$z^{2} + \frac{5 z}{3} + {(\frac{5}{6})}^{2} = \frac{2}{3} + {(\frac{5}{6})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{5}{6}$ .

$z^{2} + \frac{5 z}{3} + \frac{5^{2}}{6^{2}} = \frac{2}{3} + {(\frac{5}{6})}^{2}$

Raise $5$ to the power of $2$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{6^{2}} = \frac{2}{3} + {(\frac{5}{6})}^{2}$

Raise $6$ to the power of $2$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2}{3} + {(\frac{5}{6})}^{2}$

Simplify the right side.

Simplify $\frac{2}{3} + {(\frac{5}{6})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{5}{6}$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2}{3} + \frac{5^{2}}{6^{2}}$

Raise $5$ to the power of $2$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2}{3} + \frac{25}{6^{2}}$

Raise $6$ to the power of $2$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2}{3} + \frac{25}{36}$

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2}{3} \cdot \frac{12}{12} + \frac{25}{36}$

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{12}{12}$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2 \cdot 12}{3 \cdot 12} + \frac{25}{36}$

Multiply $3$ by $12$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2 \cdot 12}{36} + \frac{25}{36}$

Combine the numerators over the common denominator.

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{2 \cdot 12 + 25}{36}$

Simplify the numerator.

Multiply $2$ by $12$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{24 + 25}{36}$

Add $24$ and $25$ .

$z^{2} + \frac{5 z}{3} + \frac{25}{36} = \frac{49}{36}$

Step 6

Factor the perfect trinomial square into ${(z + \frac{5}{6})}^{2}$ .

${(z + \frac{5}{6})}^{2} = \frac{49}{36}$

Step 7

Solve the equation for $z$ .

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

$z + \frac{5}{6} = \pm \sqrt{\frac{49}{36}}$

Simplify $\pm \sqrt{\frac{49}{36}}$ .

Rewrite $\sqrt{\frac{49}{36}}$ as $\frac{\sqrt{49}}{\sqrt{36}}$ .

$z + \frac{5}{6} = \pm \frac{\sqrt{49}}{\sqrt{36}}$

Simplify the numerator.

Rewrite $49$ as $7^{2}$ .

$z + \frac{5}{6} = \pm \frac{\sqrt{7^{2}}}{\sqrt{36}}$

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

$z + \frac{5}{6} = \pm \frac{7}{\sqrt{36}}$

Simplify the denominator.

Rewrite $36$ as $6^{2}$ .

$z + \frac{5}{6} = \pm \frac{7}{\sqrt{6^{2}}}$

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

$z + \frac{5}{6} = \pm \frac{7}{6}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$z + \frac{5}{6} = \frac{7}{6}$

Move all terms not containing $z$ to the right side of the equation.

Subtract $\frac{5}{6}$ from both sides of the equation.

$z = \frac{7}{6} - \frac{5}{6}$

Combine the numerators over the common denominator.

$z = \frac{7 - 5}{6}$

Subtract $5$ from $7$ .

$z = \frac{2}{6}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$z = \frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Next, use the negative value of the $\pm$ to find the second solution.

$z + \frac{5}{6} = - \frac{7}{6}$

Move all terms not containing $z$ to the right side of the equation.

Subtract $\frac{5}{6}$ from both sides of the equation.

$z = - \frac{7}{6} - \frac{5}{6}$

Combine the numerators over the common denominator.

$z = \frac{- 7 - 5}{6}$

Subtract $5$ from $- 7$ .

$z = \frac{- 12}{6}$

Divide $- 12$ by $6$ .

$z = - 2$

The complete solution is the result of both the positive and negative portions of the solution.

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

Solve by Completing the Square 3z^2+5z-2=0

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