Solve for x (5x-2)^2+35x-14=-12

${(5 x - 2)}^{2} + 35 x - 14 = - 12$

Add $12$ to both sides of the equation.

${(5 x - 2)}^{2} + 35 x - 14 + 12 = 0$

Simplify ${(5 x - 2)}^{2} + 35 x - 14 + 12$ .

Simplify each term.

Rewrite ${(5 x - 2)}^{2}$ as $(5 x - 2) (5 x - 2)$ .

$(5 x - 2) (5 x - 2) + 35 x - 14 + 12 = 0$

Expand $(5 x - 2) (5 x - 2)$ using the FOIL Method.

Apply the distributive property.

$5 x (5 x - 2) - 2 (5 x - 2) + 35 x - 14 + 12 = 0$

Apply the distributive property.

$5 x (5 x) + 5 x \cdot - 2 - 2 (5 x - 2) + 35 x - 14 + 12 = 0$

Apply the distributive property.

$5 x (5 x) + 5 x \cdot - 2 - 2 (5 x) - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$5 \cdot 5 x \cdot x + 5 x \cdot - 2 - 2 (5 x) - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

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

Move $x$ .

$5 \cdot 5 (x \cdot x) + 5 x \cdot - 2 - 2 (5 x) - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

Multiply $x$ by $x$ .

$5 \cdot 5 x^{2} + 5 x \cdot - 2 - 2 (5 x) - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

Multiply $5$ by $5$ .

$25 x^{2} + 5 x \cdot - 2 - 2 (5 x) - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

Multiply $- 2$ by $5$ .

$25 x^{2} - 10 x - 2 (5 x) - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

Multiply $5$ by $- 2$ .

$25 x^{2} - 10 x - 10 x - 2 \cdot - 2 + 35 x - 14 + 12 = 0$

Multiply $- 2$ by $- 2$ .

$25 x^{2} - 10 x - 10 x + 4 + 35 x - 14 + 12 = 0$

Subtract $10 x$ from $- 10 x$ .

$25 x^{2} - 20 x + 4 + 35 x - 14 + 12 = 0$

Add $- 20 x$ and $35 x$ .

$25 x^{2} + 15 x + 4 - 14 + 12 = 0$

Subtract $14$ from $4$ .

$25 x^{2} + 15 x - 10 + 12 = 0$

Add $- 10$ and $12$ .

$25 x^{2} + 15 x + 2 = 0$

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 $a \cdot c = 25 \cdot 2 = 50$ and whose sum is $b = 15$ .

Factor $15$ out of $15 x$ .

$25 x^{2} + 15 (x) + 2 = 0$

Rewrite $15$ as $5$ plus $10$

$25 x^{2} + (5 + 10) x + 2 = 0$

Apply the distributive property.

$25 x^{2} + 5 x + 10 x + 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(25 x^{2} + 5 x) + 10 x + 2 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

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

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

$5 x + 1 = 0$

$5 x + 2 = 0$

Set $5 x + 1$ equal to $0$ and solve for $x$ .

Set $5 x + 1$ equal to $0$ .

$5 x + 1 = 0$

Solve $5 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Divide each term in $5 x = - 1$ by $5$ and simplify.

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

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 $x$ .

Subtract $2$ from 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 $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.2, - 0.4$

Solve for x (5x-2)^2+35x-14=-12

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