Solve by Factoring

$6 x^{2} - 13 x - 5 = 0$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = - 13$ .

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Step 1.1.1

Factor $- 13$ out of $- 13 x$ .

$6 x^{2} - 13 x - 5 = 0$

Step 1.1.2

Rewrite $- 13$ as $2$ plus $- 15$

$6 x^{2} + (2 - 15) x - 5 = 0$

Step 1.1.3

Apply the distributive property.

$6 x^{2} + 2 x - 15 x - 5 = 0$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$(6 x^{2} + 2 x) - 15 x - 5 = 0$

Step 1.2.2

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

$2 x (3 x + 1) - 5 (3 x + 1) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$(3 x + 1) (2 x - 5) = 0$

Step 2

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

$3 x + 1 = 0$

$2 x - 5 = 0$

Step 3

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 3.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 3.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 4.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 4.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 4.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 5

The final solution is all the values that make $(3 x + 1) (2 x - 5) = 0$ true.

$x = - \frac{1}{3}, \frac{5}{2}$

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