Solve Using the Quadratic Formula (z^2)/3=z/2+5/6

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{z}{2}$ from both sides of the equation.

$\frac{z^{2}}{3} - \frac{z}{2} = \frac{5}{6}$

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

$\frac{z^{2}}{3} - \frac{z}{2} - \frac{5}{6} = 0$

Step 2

Multiply through by the least common denominator $6$ , then simplify.

Apply the distributive property.

$6 (\frac{z^{2}}{3}) + 6 (- \frac{z}{2}) + 6 (- \frac{5}{6}) = 0$

Simplify.

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$3 (2) (\frac{z^{2}}{3}) + 6 (- \frac{z}{2}) + 6 (- \frac{5}{6}) = 0$

Cancel the common factor.

$3 \cdot (2 (\frac{z^{2}}{3})) + 6 (- \frac{z}{2}) + 6 (- \frac{5}{6}) = 0$

Rewrite the expression.

$2 z^{2} + 6 (- \frac{z}{2}) + 6 (- \frac{5}{6}) = 0$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{z}{2}$ into the numerator.

$2 z^{2} + 6 (\frac{- z}{2}) + 6 (- \frac{5}{6}) = 0$

Factor $2$ out of $6$ .

$2 z^{2} + 2 (3) (\frac{- z}{2}) + 6 (- \frac{5}{6}) = 0$

Cancel the common factor.

$2 z^{2} + 2 \cdot (3 (\frac{- z}{2})) + 6 (- \frac{5}{6}) = 0$

Rewrite the expression.

$2 z^{2} + 3 (- z) + 6 (- \frac{5}{6}) = 0$

Multiply $- 1$ by $3$ .

$2 z^{2} - 3 z + 6 (- \frac{5}{6}) = 0$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$2 z^{2} - 3 z + 6 (\frac{- 5}{6}) = 0$

Cancel the common factor.

$2 z^{2} - 3 z + 6 (\frac{- 5}{6}) = 0$

Rewrite the expression.

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = - 3$ , and $c = - 5$ into the quadratic formula and solve for $z$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot - 5)}}{2 \cdot 2}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$z = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 5$ .

Multiply $- 4$ by $2$ .

$z = \frac{3 \pm \sqrt{9 - 8 \cdot - 5}}{2 \cdot 2}$

Multiply $- 8$ by $- 5$ .

$z = \frac{3 \pm \sqrt{9 + 40}}{2 \cdot 2}$

Add $9$ and $40$ .

$z = \frac{3 \pm \sqrt{49}}{2 \cdot 2}$

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

$z = \frac{3 \pm \sqrt{7^{2}}}{2 \cdot 2}$

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

$z = \frac{3 \pm 7}{2 \cdot 2}$

Multiply $2$ by $2$ .

$z = \frac{3 \pm 7}{4}$

Step 6

The final answer is the combination of both solutions.

$z = \frac{5}{2}, - 1$

Solve Using the Quadratic Formula (z^2)/3=z/2+5/6

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