Solve by Factoring x(15x+1)=6

$x (15 x + 1) = 6$

Step 1

Subtract $6$ from both sides of the equation.

$x (15 x + 1) - 6 = 0$

Step 2

Simplify each term.

Apply the distributive property.

$x (15 x) + x \cdot 1 - 6 = 0$

Rewrite using the commutative property of multiplication.

$15 x \cdot x + x \cdot 1 - 6 = 0$

Multiply $x$ by $1$ .

$15 x \cdot x + x - 6 = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$15 (x \cdot x) + x - 6 = 0$

Multiply $x$ by $x$ .

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

Step 3

Factor by grouping.

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 $a \cdot c = 15 \cdot - 6 = - 90$ and whose sum is $b = 1$ .

Multiply by $1$ .

$15 x^{2} + 1 x - 6 = 0$

Rewrite $1$ as $- 9$ plus $10$

$15 x^{2} + (- 9 + 10) x - 6 = 0$

Apply the distributive property.

$15 x^{2} - 9 x + 10 x - 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(15 x^{2} - 9 x) + 10 x - 6 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x - 3$ .

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

Step 4

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

$5 x - 3 = 0$

$3 x + 2 = 0$

Step 5

Set $5 x - 3$ equal to $0$ and solve for $x$ .

Set $5 x - 3$ equal to $0$ .

$5 x - 3 = 0$

Solve $5 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$5 x = 3$

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

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 6

Set $3 x + 2$ equal to $0$ and solve for $x$ .

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Solve $3 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Divide each term in $3 x = - 2$ by $3$ and simplify.

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 7

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

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

Solve by Factoring x(15x+1)=6

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