Basic Math Examples

$(18 + 2 v) (24 + 2 v) = 396$

Step 1

Simplify $(18 + 2 v) (24 + 2 v)$ .

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Step 1.1

Expand $(18 + 2 v) (24 + 2 v)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$18 (24 + 2 v) + 2 v (24 + 2 v) = 396$

Step 1.1.2

Apply the distributive property.

$18 \cdot 24 + 18 (2 v) + 2 v (24 + 2 v) = 396$

Step 1.1.3

Apply the distributive property.

$18 \cdot 24 + 18 (2 v) + 2 v \cdot 24 + 2 v (2 v) = 396$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $18$ by $24$ .

$432 + 18 (2 v) + 2 v \cdot 24 + 2 v (2 v) = 396$

Step 1.2.1.2

Multiply $2$ by $18$ .

$432 + 36 v + 2 v \cdot 24 + 2 v (2 v) = 396$

Step 1.2.1.3

Multiply $24$ by $2$ .

$432 + 36 v + 48 v + 2 v (2 v) = 396$

Step 1.2.1.4

Rewrite using the commutative property of multiplication.

$432 + 36 v + 48 v + 2 \cdot 2 v \cdot v = 396$

Step 1.2.1.5

Multiply $v$ by $v$ by adding the exponents.

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Step 1.2.1.5.1

Move $v$ .

$432 + 36 v + 48 v + 2 \cdot 2 (v \cdot v) = 396$

Step 1.2.1.5.2

Multiply $v$ by $v$ .

$432 + 36 v + 48 v + 2 \cdot 2 v^{2} = 396$

Step 1.2.1.6

Multiply $2$ by $2$ .

$432 + 36 v + 48 v + 4 v^{2} = 396$

Step 1.2.2

Add $36 v$ and $48 v$ .

$432 + 84 v + 4 v^{2} = 396$

Step 2

Subtract $396$ from both sides of the equation.

$432 + 84 v + 4 v^{2} - 396 = 0$

Step 3

Subtract $396$ from $432$ .

$84 v + 4 v^{2} + 36 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $4$ out of $84 v + 4 v^{2} + 36$ .

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Step 4.1.1

Factor $4$ out of $84 v$ .

$4 (21 v) + 4 v^{2} + 36 = 0$

Step 4.1.2

Factor $4$ out of $4 v^{2}$ .

$4 (21 v) + 4 (v^{2}) + 36 = 0$

Step 4.1.3

Factor $4$ out of $36$ .

$4 (21 v) + 4 v^{2} + 4 \cdot 9 = 0$

Step 4.1.4

Factor $4$ out of $4 (21 v) + 4 v^{2}$ .

$4 (21 v + v^{2}) + 4 \cdot 9 = 0$

Step 4.1.5

Factor $4$ out of $4 (21 v + v^{2}) + 4 \cdot 9$ .

$4 (21 v + v^{2} + 9) = 0$

Step 4.2

Reorder terms.

$4 (v^{2} + 21 v + 9) = 0$

Step 5

Divide each term in $4 (v^{2} + 21 v + 9) = 0$ by $4$ and simplify.

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Step 5.1

Divide each term in $4 (v^{2} + 21 v + 9) = 0$ by $4$ .

$\frac{4 (v^{2} + 21 v + 9)}{4} = \frac{0}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 (v^{2} + 21 v + 9)}{4} = \frac{0}{4}$

Step 5.2.1.2

Divide $v^{2} + 21 v + 9$ by $1$ .

$v^{2} + 21 v + 9 = \frac{0}{4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $4$ .

$v^{2} + 21 v + 9 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = 1$ , $b = 21$ , and $c = 9$ into the quadratic formula and solve for $v$ .

$\frac{- 21 \pm \sqrt{21^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $21$ to the power of $2$ .

$v = \frac{- 21 \pm \sqrt{441 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$v = \frac{- 21 \pm \sqrt{441 - 4 \cdot 9}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $9$ .

$v = \frac{- 21 \pm \sqrt{441 - 36}}{2 \cdot 1}$

Step 8.1.3

Subtract $36$ from $441$ .

$v = \frac{- 21 \pm \sqrt{405}}{2 \cdot 1}$

Step 8.1.4

Rewrite $405$ as $9^{2} \cdot 5$ .

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Step 8.1.4.1

Factor $81$ out of $405$ .

$v = \frac{- 21 \pm \sqrt{81 (5)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $81$ as $9^{2}$ .

$v = \frac{- 21 \pm \sqrt{9^{2} \cdot 5}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$v = \frac{- 21 \pm 9 \sqrt{5}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$v = \frac{- 21 \pm 9 \sqrt{5}}{2}$

Step 9

The final answer is the combination of both solutions.

$v = - \frac{21 - 9 \sqrt{5}}{2}, - \frac{21 + 9 \sqrt{5}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{21 - 9 \sqrt{5}}{2}, - \frac{21 + 9 \sqrt{5}}{2}$

Decimal Form:

$v = - 0.43769410 \dots, - 20.56230589 \dots$