Solve Using the Quadratic Formula 5/6k^2-k-12/5=0

$\frac{5}{6} k^{2} - k - \frac{12}{5} = 0$

Step 1

Simplify the left side.

Combine $\frac{5}{6}$ and $k^{2}$ .

$\frac{5 k^{2}}{6} - k - \frac{12}{5} = 0$

Step 2

Multiply through by the least common denominator $30$ , then simplify.

Apply the distributive property.

$30 (\frac{5 k^{2}}{6}) + 30 (- k) + 30 (- \frac{12}{5}) = 0$

Simplify.

Cancel the common factor of $6$ .

Factor $6$ out of $30$ .

$6 (5) (\frac{5 k^{2}}{6}) + 30 (- k) + 30 (- \frac{12}{5}) = 0$

Cancel the common factor.

$6 \cdot (5 (\frac{5 k^{2}}{6})) + 30 (- k) + 30 (- \frac{12}{5}) = 0$

Rewrite the expression.

$5 (5 k^{2}) + 30 (- k) + 30 (- \frac{12}{5}) = 0$

Multiply $5$ by $5$ .

$25 k^{2} + 30 (- k) + 30 (- \frac{12}{5}) = 0$

Multiply $- 1$ by $30$ .

$25 k^{2} - 30 k + 30 (- \frac{12}{5}) = 0$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{12}{5}$ into the numerator.

$25 k^{2} - 30 k + 30 (\frac{- 12}{5}) = 0$

Factor $5$ out of $30$ .

$25 k^{2} - 30 k + 5 (6) (\frac{- 12}{5}) = 0$

Cancel the common factor.

$25 k^{2} - 30 k + 5 \cdot (6 (\frac{- 12}{5})) = 0$

Rewrite the expression.

$25 k^{2} - 30 k + 6 \cdot - 12 = 0$

Multiply $6$ by $- 12$ .

$25 k^{2} - 30 k - 72 = 0$

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 $a = 25$ , $b = - 30$ , and $c = - 72$ into the quadratic formula and solve for $k$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (25 \cdot - 72)}}{2 \cdot 25}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$k = \frac{30 \pm \sqrt{900 - 4 \cdot 25 \cdot - 72}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot - 72$ .

Multiply $- 4$ by $25$ .

$k = \frac{30 \pm \sqrt{900 - 100 \cdot - 72}}{2 \cdot 25}$

Multiply $- 100$ by $- 72$ .

$k = \frac{30 \pm \sqrt{900 + 7200}}{2 \cdot 25}$

Add $900$ and $7200$ .

$k = \frac{30 \pm \sqrt{8100}}{2 \cdot 25}$

Rewrite $8100$ as $90^{2}$ .

$k = \frac{30 \pm \sqrt{90^{2}}}{2 \cdot 25}$

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

$k = \frac{30 \pm 90}{2 \cdot 25}$

Multiply $2$ by $25$ .

$k = \frac{30 \pm 90}{50}$

Simplify $\frac{30 \pm 90}{50}$ .

$k = \frac{3 \pm 9}{5}$

Step 6

The final answer is the combination of both solutions.

$k = \frac{12}{5}, - \frac{6}{5}$

Solve Using the Quadratic Formula 5/6k^2-k-12/5=0

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