Precalculus Examples

$\frac{3}{5} k^{2} - k - \frac{10}{3} = 0$

Step 1

Combine $\frac{3}{5}$ and $k^{2}$ .

$\frac{3 k^{2}}{5} - k - \frac{10}{3} = 0$

Step 2

Multiply each term in $\frac{3 k^{2}}{5} - k - \frac{10}{3} = 0$ by $15$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{3 k^{2}}{5} - k - \frac{10}{3} = 0$ by $15$ .

$\frac{3 k^{2}}{5} \cdot 15 - k \cdot 15 - \frac{10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.1

Factor $5$ out of $15$ .

$\frac{3 k^{2}}{5} \cdot (5 (3)) - k \cdot 15 - \frac{10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{3 k^{2}}{5} \cdot (5 \cdot 3) - k \cdot 15 - \frac{10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2.1.1.3

Rewrite the expression.

$3 k^{2} \cdot 3 - k \cdot 15 - \frac{10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2.1.2

Multiply $3$ by $3$ .

$9 k^{2} - k \cdot 15 - \frac{10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2.1.3

Multiply $15$ by $- 1$ .

$9 k^{2} - 15 k - \frac{10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.4.1

Move the leading negative in $- \frac{10}{3}$ into the numerator.

$9 k^{2} - 15 k + \frac{- 10}{3} \cdot 15 = 0 \cdot 15$

Step 2.2.1.4.2

Factor $3$ out of $15$ .

$9 k^{2} - 15 k + \frac{- 10}{3} \cdot (3 (5)) = 0 \cdot 15$

Step 2.2.1.4.3

Cancel the common factor.

$9 k^{2} - 15 k + \frac{- 10}{3} \cdot (3 \cdot 5) = 0 \cdot 15$

Step 2.2.1.4.4

Rewrite the expression.

$9 k^{2} - 15 k - 10 \cdot 5 = 0 \cdot 15$

Step 2.2.1.5

Multiply $- 10$ by $5$ .

$9 k^{2} - 15 k - 50 = 0 \cdot 15$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $15$ .

$9 k^{2} - 15 k - 50 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 9 \cdot - 50 = - 450$ and whose sum is $b = - 15$ .

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Step 3.1.1

Factor $- 15$ out of $- 15 k$ .

$9 k^{2} - 15 k - 50 = 0$

Step 3.1.2

Rewrite $- 15$ as $15$ plus $- 30$

$9 k^{2} + (15 - 30) k - 50 = 0$

Step 3.1.3

Apply the distributive property.

$9 k^{2} + 15 k - 30 k - 50 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(9 k^{2} + 15 k) - 30 k - 50 = 0$

Step 3.2.2

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

$3 k (3 k + 5) - 10 (3 k + 5) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 k + 5$ .

$(3 k + 5) (3 k - 10) = 0$

Step 4

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

$3 k + 5 = 0$

$3 k - 10 = 0$

Step 5

Set $3 k + 5$ equal to $0$ and solve for $k$ .

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Step 5.1

Set $3 k + 5$ equal to $0$ .

$3 k + 5 = 0$

Step 5.2

Solve $3 k + 5 = 0$ for $k$ .

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Step 5.2.1

Subtract $5$ from both sides of the equation.

$3 k = - 5$

Step 5.2.2

Divide each term in $3 k = - 5$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 k = - 5$ by $3$ .

$\frac{3 k}{3} = \frac{- 5}{3}$

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.

$\frac{3 k}{3} = \frac{- 5}{3}$

Step 5.2.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 5}{3}$

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.

$k = - \frac{5}{3}$

Step 6

Set $3 k - 10$ equal to $0$ and solve for $k$ .

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Step 6.1

Set $3 k - 10$ equal to $0$ .

$3 k - 10 = 0$

Step 6.2

Solve $3 k - 10 = 0$ for $k$ .

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Step 6.2.1

Add $10$ to both sides of the equation.

$3 k = 10$

Step 6.2.2

Divide each term in $3 k = 10$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 k = 10$ by $3$ .

$\frac{3 k}{3} = \frac{10}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 k}{3} = \frac{10}{3}$

Step 6.2.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{10}{3}$

Step 7

The final solution is all the values that make $(3 k + 5) (3 k - 10) = 0$ true.

$k = - \frac{5}{3}, \frac{10}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{5}{3}, \frac{10}{3}$

Decimal Form:

$k = - 1.\overline{6}, 3.\overline{3}$

Mixed Number Form:

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