Pre-Algebra Examples

$x (5 + 12 x) = 28$

Step 1

Simplify $x (5 + 12 x)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x \cdot 5 + x (12 x) = 28$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $5$ to the left of $x$ .

$5 \cdot x + x (12 x) = 28$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$5 \cdot x + 12 x \cdot x = 28$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$5 x + 12 (x \cdot x) = 28$

Step 1.2.2

Multiply $x$ by $x$ .

$5 x + 12 x^{2} = 28$

Step 2

Subtract $28$ from both sides of the equation.

$5 x + 12 x^{2} - 28 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$5 u + 12 u^{2} - 28 = 0$

Step 3.2

Factor by grouping.

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Step 3.2.1

Reorder terms.

$12 u^{2} + 5 u - 28 = 0$

Step 3.2.2

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 $a \cdot c = 12 \cdot - 28 = - 336$ and whose sum is $b = 5$ .

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Step 3.2.2.1

Factor $5$ out of $5 u$ .

$12 u^{2} + 5 (u) - 28 = 0$

Step 3.2.2.2

Rewrite $5$ as $- 16$ plus $21$

$12 u^{2} + (- 16 + 21) u - 28 = 0$

Step 3.2.2.3

Apply the distributive property.

$12 u^{2} - 16 u + 21 u - 28 = 0$

Step 3.2.3

Factor out the greatest common factor from each group.

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Step 3.2.3.1

Group the first two terms and the last two terms.

$(12 u^{2} - 16 u) + 21 u - 28 = 0$

Step 3.2.3.2

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

$4 u (3 u - 4) + 7 (3 u - 4) = 0$

Step 3.2.4

Factor the polynomial by factoring out the greatest common factor, $3 u - 4$ .

$(3 u - 4) (4 u + 7) = 0$

Step 3.3

Replace all occurrences of $u$ with $x$ .

$(3 x - 4) (4 x + 7) = 0$

Step 4

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

$3 x - 4 = 0$

$4 x + 7 = 0$

Step 5

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 5.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 5.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 5.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 6

Set $4 x + 7$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $4 x + 7$ equal to $0$ .

$4 x + 7 = 0$

Step 6.2

Solve $4 x + 7 = 0$ for $x$ .

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Step 6.2.1

Subtract $7$ from both sides of the equation.

$4 x = - 7$

Step 6.2.2

Divide each term in $4 x = - 7$ by $4$ and simplify.

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Step 6.2.2.1

Divide each term in $4 x = - 7$ by $4$ .

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{4}$

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.

$x = - \frac{7}{4}$

Step 7

The final solution is all the values that make $(3 x - 4) (4 x + 7) = 0$ true.

$x = \frac{4}{3}, - \frac{7}{4}$