Algebra Examples

$3 (3 x - 2) = (x + 4) (4 - x)$

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.

$(x + 4) (4 - x) = 3 (3 x - 2)$

Step 2

Simplify $(x + 4) (4 - x)$ .

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Step 2.1

Rewrite.

$0 + 0 + (x + 4) (4 - x) = 3 (3 x - 2)$

Step 2.2

Simplify by adding zeros.

$(x + 4) (4 - x) = 3 (3 x - 2)$

Step 2.3

Expand $(x + 4) (4 - x)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$x (4 - x) + 4 (4 - x) = 3 (3 x - 2)$

Step 2.3.2

Apply the distributive property.

$x \cdot 4 + x (- x) + 4 (4 - x) = 3 (3 x - 2)$

Step 2.3.3

Apply the distributive property.

$x \cdot 4 + x (- x) + 4 \cdot 4 + 4 (- x) = 3 (3 x - 2)$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Move $4$ to the left of $x$ .

$4 \cdot x + x (- x) + 4 \cdot 4 + 4 (- x) = 3 (3 x - 2)$

Step 2.4.1.2

Rewrite using the commutative property of multiplication.

$4 x - x \cdot x + 4 \cdot 4 + 4 (- x) = 3 (3 x - 2)$

Step 2.4.1.3

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

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Step 2.4.1.3.1

Move $x$ .

$4 x - (x \cdot x) + 4 \cdot 4 + 4 (- x) = 3 (3 x - 2)$

Step 2.4.1.3.2

Multiply $x$ by $x$ .

$4 x - x^{2} + 4 \cdot 4 + 4 (- x) = 3 (3 x - 2)$

Step 2.4.1.4

Multiply $4$ by $4$ .

$4 x - x^{2} + 16 + 4 (- x) = 3 (3 x - 2)$

Step 2.4.1.5

Multiply $- 1$ by $4$ .

$4 x - x^{2} + 16 - 4 x = 3 (3 x - 2)$

Step 2.4.2

Subtract $4 x$ from $4 x$ .

$- x^{2} + 16 + 0 = 3 (3 x - 2)$

Step 2.4.3

Add $- x^{2}$ and $0$ .

$- x^{2} + 16 = 3 (3 x - 2)$

Step 3

Simplify $3 (3 x - 2)$ .

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Step 3.1

Apply the distributive property.

$- x^{2} + 16 = 3 (3 x) + 3 \cdot - 2$

Step 3.2

Multiply.

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Step 3.2.1

Multiply $3$ by $3$ .

$- x^{2} + 16 = 9 x + 3 \cdot - 2$

Step 3.2.2

Multiply $3$ by $- 2$ .

$- x^{2} + 16 = 9 x - 6$

Step 4

Subtract $9 x$ from both sides of the equation.

$- x^{2} + 16 - 9 x = - 6$

Step 5

Add $6$ to both sides of the equation.

$- x^{2} + 16 - 9 x + 6 = 0$

Step 6

Add $16$ and $6$ .

$- x^{2} - 9 x + 22 = 0$

Step 7

Factor the left side of the equation.

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Step 7.1

Factor $- 1$ out of $- x^{2} - 9 x + 22$ .

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Step 7.1.1

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

$- (x^{2}) - 9 x + 22 = 0$

Step 7.1.2

Factor $- 1$ out of $- 9 x$ .

$- (x^{2}) - (9 x) + 22 = 0$

Step 7.1.3

Rewrite $22$ as $- 1 (- 22)$ .

$- (x^{2}) - (9 x) - 1 \cdot - 22 = 0$

Step 7.1.4

Factor $- 1$ out of $- (x^{2}) - (9 x)$ .

$- (x^{2} + 9 x) - 1 \cdot - 22 = 0$

Step 7.1.5

Factor $- 1$ out of $- (x^{2} + 9 x) - 1 (- 22)$ .

$- (x^{2} + 9 x - 22) = 0$

Step 7.2

Factor.

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Step 7.2.1

Factor $x^{2} + 9 x - 22$ using the AC method.

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Step 7.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 22$ and whose sum is $9$ .

$- 2, 11$

Step 7.2.1.2

Write the factored form using these integers.

$- ((x - 2) (x + 11)) = 0$

Step 7.2.2

Remove unnecessary parentheses.

$- (x - 2) (x + 11) = 0$

Step 8

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

$x - 2 = 0$

$x + 11 = 0$

Step 9

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

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Step 9.1

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

$x - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$x = 2$

Step 10

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Step 10.2

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 11

The final solution is all the values that make $- (x - 2) (x + 11) = 0$ true.

$x = 2, - 11$