Solve for x x(5x+13)=6

$x (5 x + 13) = 6$

Step 1

Simplify $x (5 x + 13)$ .

Simplify by multiplying through.

Apply the distributive property.

$x (5 x) + x \cdot 13 = 6$

Reorder.

Rewrite using the commutative property of multiplication.

$5 x \cdot x + x \cdot 13 = 6$

Move $13$ to the left of $x$ .

$5 x \cdot x + 13 \cdot x = 6$

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

Move $x$ .

$5 (x \cdot x) + 13 \cdot x = 6$

Multiply $x$ by $x$ .

$5 x^{2} + 13 \cdot x = 6$

$5 x^{2} + 13 x = 6$

Step 2

Subtract $6$ from both sides of the equation.

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

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

Factor $13$ out of $13 x$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

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

Step 4

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

$5 x - 2 = 0$

$x + 3 = 0$

Step 5

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

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

$5 x - 2 = 0$

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

Add $2$ to both sides of the equation.

$5 x = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 6

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.4, - 3$

Solve for x x(5x+13)=6

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