Pre-Algebra Examples

$(3 x + 5) (x - 5) = - 2 x (6 x + 5) + x - 19$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 2 x (6 x + 5) + x - 19 = (3 x + 5) (x - 5)$

Step 2

Simplify $- 2 x (6 x + 5) + x - 19$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- 2 x (6 x) - 2 x \cdot 5 + x - 19 = (3 x + 5) (x - 5)$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$- 2 \cdot 6 x \cdot x - 2 x \cdot 5 + x - 19 = (3 x + 5) (x - 5)$

Step 2.1.3

Multiply $5$ by $- 2$ .

$- 2 \cdot 6 x \cdot x - 10 x + x - 19 = (3 x + 5) (x - 5)$

Step 2.1.4

Simplify each term.

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Step 2.1.4.1

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

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Step 2.1.4.1.1

Move $x$ .

$- 2 \cdot 6 (x \cdot x) - 10 x + x - 19 = (3 x + 5) (x - 5)$

Step 2.1.4.1.2

Multiply $x$ by $x$ .

$- 2 \cdot 6 x^{2} - 10 x + x - 19 = (3 x + 5) (x - 5)$

Step 2.1.4.2

Multiply $- 2$ by $6$ .

$- 12 x^{2} - 10 x + x - 19 = (3 x + 5) (x - 5)$

Step 2.2

Add $- 10 x$ and $x$ .

$- 12 x^{2} - 9 x - 19 = (3 x + 5) (x - 5)$

Step 3

Simplify $(3 x + 5) (x - 5)$ .

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Step 3.1

Expand $(3 x + 5) (x - 5)$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$- 12 x^{2} - 9 x - 19 = 3 x (x - 5) + 5 (x - 5)$

Step 3.1.2

Apply the distributive property.

$- 12 x^{2} - 9 x - 19 = 3 x \cdot x + 3 x \cdot - 5 + 5 (x - 5)$

Step 3.1.3

Apply the distributive property.

$- 12 x^{2} - 9 x - 19 = 3 x \cdot x + 3 x \cdot - 5 + 5 x + 5 \cdot - 5$

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Move $x$ .

$- 12 x^{2} - 9 x - 19 = 3 (x \cdot x) + 3 x \cdot - 5 + 5 x + 5 \cdot - 5$

Step 3.2.1.1.2

Multiply $x$ by $x$ .

$- 12 x^{2} - 9 x - 19 = 3 x^{2} + 3 x \cdot - 5 + 5 x + 5 \cdot - 5$

Step 3.2.1.2

Multiply $- 5$ by $3$ .

$- 12 x^{2} - 9 x - 19 = 3 x^{2} - 15 x + 5 x + 5 \cdot - 5$

Step 3.2.1.3

Multiply $5$ by $- 5$ .

$- 12 x^{2} - 9 x - 19 = 3 x^{2} - 15 x + 5 x - 25$

Step 3.2.2

Add $- 15 x$ and $5 x$ .

$- 12 x^{2} - 9 x - 19 = 3 x^{2} - 10 x - 25$

Step 4

Move all terms containing $x$ to the left side of the equation.

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Step 4.1

Subtract $3 x^{2}$ from both sides of the equation.

$- 12 x^{2} - 9 x - 19 - 3 x^{2} = - 10 x - 25$

Step 4.2

Add $10 x$ to both sides of the equation.

$- 12 x^{2} - 9 x - 19 - 3 x^{2} + 10 x = - 25$

Step 4.3

Subtract $3 x^{2}$ from $- 12 x^{2}$ .

$- 15 x^{2} - 9 x - 19 + 10 x = - 25$

Step 4.4

Add $- 9 x$ and $10 x$ .

$- 15 x^{2} + x - 19 = - 25$

Step 5

Add $25$ to both sides of the equation.

$- 15 x^{2} + x - 19 + 25 = 0$

Step 6

Add $- 19$ and $25$ .

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

Step 7

Factor the left side of the equation.

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Step 7.1

Factor $- 1$ out of $- 15 x^{2} + x + 6$ .

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Step 7.1.1

Factor $- 1$ out of $- 15 x^{2}$ .

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

Step 7.1.2

Factor $- 1$ out of $x$ .

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

Step 7.1.3

Rewrite $6$ as $- 1 (- 6)$ .

$- (15 x^{2}) - 1 (- x) - 1 \cdot - 6 = 0$

Step 7.1.4

Factor $- 1$ out of $- (15 x^{2}) - 1 (- x)$ .

$- (15 x^{2} - x) - 1 \cdot - 6 = 0$

Step 7.1.5

Factor $- 1$ out of $- (15 x^{2} - x) - 1 (- 6)$ .

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

Step 7.2

Factor.

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Step 7.2.1

Factor by grouping.

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Step 7.2.1.1

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

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Step 7.2.1.1.1

Factor $- 1$ out of $- x$ .

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

Step 7.2.1.1.2

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

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

Step 7.2.1.1.3

Apply the distributive property.

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

Step 7.2.1.2

Factor out the greatest common factor from each group.

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Step 7.2.1.2.1

Group the first two terms and the last two terms.

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

Step 7.2.1.2.2

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

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

Step 7.2.1.3

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

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

Step 7.2.2

Remove unnecessary parentheses.

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

Step 8

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 9

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

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Step 9.1

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

$5 x + 3 = 0$

Step 9.2

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

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Step 9.2.1

Subtract $3$ from both sides of the equation.

$5 x = - 3$

Step 9.2.2

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

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Step 9.2.2.1

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

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

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $5$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Move the negative in front of the fraction.

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

Step 10

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

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Step 10.1

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

$3 x - 2 = 0$

Step 10.2

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

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Step 10.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 10.2.2

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

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Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $3$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

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

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 11

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

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