Algebra Examples

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

Step 1

Add $12$ to both sides of the equation.

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

Apply the distributive property.

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

Step 2.1.2.3

Apply the distributive property.

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

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.3.1.2

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

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Step 2.1.3.1.2.1

Move $x$ .

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

Step 2.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3.1.3

Multiply $5$ by $5$ .

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

Step 2.1.3.1.4

Multiply $- 2$ by $5$ .

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

Step 2.1.3.1.5

Multiply $5$ by $- 2$ .

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

Step 2.1.3.1.6

Multiply $- 2$ by $- 2$ .

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

Step 2.1.3.2

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

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

Step 2.2

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

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

Step 2.3

Subtract $14$ from $4$ .

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

Step 2.4

Add $- 10$ and $12$ .

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

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

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Step 3.1.1

Factor $15$ out of $15 x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

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.

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 4

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

$5 x + 1 = 0$

$5 x + 2 = 0$

Step 5

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

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Step 5.1

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

$5 x + 1 = 0$

Step 5.2

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

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 6

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

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Step 6.1

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

$5 x + 2 = 0$

Step 6.2

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

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Step 6.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $5$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.2, - 0.4$