Basic Math Examples

$3.5 = \frac{2374.8 \cdot b + 40 \cdot b^{2}}{1.35 \cdot (720 \cdot b - 4326)}$

Step 1

Rewrite the equation as $\frac{2374.8 \cdot b + 40 \cdot b^{2}}{1.35 \cdot (720 \cdot b - 4326)} = 3.5$ .

$\frac{2374.8 \cdot b + 40 \cdot b^{2}}{1.35 \cdot (720 \cdot b - 4326)} = 3.5$

Step 2

Factor each term.

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Step 2.1

Factor $0.4 b$ out of $2374.8 \cdot b + 40 \cdot b^{2}$ .

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Step 2.1.1

Factor $0.4 b$ out of $2374.8 \cdot b$ .

$\frac{0.4 b (5937) + 40 \cdot b^{2}}{1.35 \cdot (720 \cdot b - 4326)} = 3.5$

Step 2.1.2

Factor $0.4 b$ out of $40 \cdot b^{2}$ .

$\frac{0.4 b (5937) + 0.4 b (100 \cdot b)}{1.35 \cdot (720 \cdot b - 4326)} = 3.5$

Step 2.1.3

Factor $0.4 b$ out of $0.4 b (5937) + 0.4 b (100 \cdot b)$ .

$\frac{0.4 b (5937 + 100 \cdot b)}{1.35 \cdot (720 \cdot b - 4326)} = 3.5$

$\frac{0.4 b (5937 + 100 b)}{1.35 \cdot (720 \cdot b - 4326)} = 3.5$

Step 2.2

Factor $6$ out of $720 \cdot b - 4326$ .

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Step 2.2.1

Factor $6$ out of $720 \cdot b$ .

$\frac{0.4 b (5937 + 100 b)}{1.35 \cdot (6 (120 \cdot b) - 4326)} = 3.5$

Step 2.2.2

Factor $6$ out of $- 4326$ .

$\frac{0.4 b (5937 + 100 b)}{1.35 \cdot (6 (120 \cdot b) + 6 (- 721))} = 3.5$

Step 2.2.3

Factor $6$ out of $6 (120 \cdot b) + 6 (- 721)$ .

$\frac{0.4 b (5937 + 100 b)}{1.35 \cdot (6 (120 \cdot b - 721))} = 3.5$

$\frac{0.4 b (5937 + 100 b)}{1.35 \cdot (6 (120 b - 721))} = 3.5$

Step 2.3

Remove unnecessary parentheses.

$\frac{0.4 b (5937 + 100 b)}{1.35 \cdot 6 (120 b - 721)} = 3.5$

Step 2.4

Multiply $1.35$ by $6$ .

$\frac{0.4 b (5937 + 100 b)}{8.1 (120 b - 721)} = 3.5$

Step 2.5

Factor $0.4$ out of $0.4 b (5937 + 100 b)$ .

$\frac{0.4 (b (5937 + 100 b))}{8.1 (120 b - 721)} = 3.5$

Step 2.6

Separate fractions.

$\frac{0.4}{8.1} \cdot \frac{b (5937 + 100 b)}{120 b - 721} = 3.5$

Step 2.7

Divide $0.4$ by $8.1$ .

$0.04938271 \frac{b (5937 + 100 b)}{120 b - 721} = 3.5$

Step 2.8

Combine $0.04938271$ and $\frac{b (5937 + 100 b)}{120 b - 721}$ .

$\frac{0.04938271 (b (5937 + 100 b))}{120 b - 721} = 3.5$

Step 2.9

Remove parentheses.

$\frac{0.04938271 b (5937 + 100 b)}{120 b - 721} = 3.5$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$120 b - 721, 1$

Step 3.2

Remove parentheses.

$120 b - 721, 1$

Step 3.3

The LCM of one and any expression is the expression.

$120 b - 721$

Step 4

Multiply each term in $\frac{0.04938271 b (5937 + 100 b)}{120 b - 721} = 3.5$ by $120 b - 721$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{0.04938271 b (5937 + 100 b)}{120 b - 721} = 3.5$ by $120 b - 721$ .

$\frac{0.04938271 b (5937 + 100 b)}{120 b - 721} (120 b - 721) = 3.5 (120 b - 721)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify terms.

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Step 4.2.1.1

Cancel the common factor of $120 b - 721$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{0.04938271 b (5937 + 100 b)}{120 b - 721} (120 b - 721) = 3.5 (120 b - 721)$

Step 4.2.1.1.2

Rewrite the expression.

$0.04938271 b (5937 + 100 b) = 3.5 (120 b - 721)$

Step 4.2.1.2

Apply the distributive property.

$0.04938271 b \cdot 5937 + 0.04938271 b (100 b) = 3.5 (120 b - 721)$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Multiply $5937$ by $0.04938271$ .

$293.18518518 b + 0.04938271 b (100 b) = 3.5 (120 b - 721)$

Step 4.2.1.3.2

Rewrite using the commutative property of multiplication.

$293.18518518 b + 0.04938271 \cdot 100 b \cdot b = 3.5 (120 b - 721)$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Multiply $b$ by $b$ by adding the exponents.

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Step 4.2.2.1.1

Move $b$ .

$293.18518518 b + 0.04938271 \cdot 100 (b \cdot b) = 3.5 (120 b - 721)$

Step 4.2.2.1.2

Multiply $b$ by $b$ .

$293.18518518 b + 0.04938271 \cdot 100 b^{2} = 3.5 (120 b - 721)$

Step 4.2.2.2

Multiply $0.04938271$ by $100$ .

$293.18518518 b + 4.9382716 b^{2} = 3.5 (120 b - 721)$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$293.18518518 b + 4.9382716 b^{2} = 3.5 (120 b) + 3.5 \cdot - 721$

Step 4.3.2

Multiply.

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Step 4.3.2.1

Multiply $120$ by $3.5$ .

$293.18518518 b + 4.9382716 b^{2} = 420 b + 3.5 \cdot - 721$

Step 4.3.2.2

Multiply $3.5$ by $- 721$ .

$293.18518518 b + 4.9382716 b^{2} = 420 b - 2523.5$

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

Subtract $420 b$ from both sides of the equation.

$293.18518518 b + 4.9382716 b^{2} - 420 b = - 2523.5$

Step 5.1.2

Subtract $420 b$ from $293.18518518 b$ .

$4.9382716 b^{2} - 126.81481481 b = - 2523.5$

Step 5.2

Add $2523.5$ to both sides of the equation.

$4.9382716 b^{2} - 126.81481481 b + 2523.5 = 0$

Step 5.3

Use the quadratic formula to find the solutions.

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

Step 5.4

Substitute the values $a = 4.9382716$ , $b = - 126.81481481$ , and $c = 2523.5$ into the quadratic formula and solve for $b$ .

$\frac{126.81481481 \pm \sqrt{{(- 126.81481481)}^{2} - 4 \cdot (4.9382716 \cdot 2523.5)}}{2 \cdot 4.9382716}$

Step 5.5

Simplify.

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Raise $- 126.81481481$ to the power of $2$ .

$b = \frac{126.81481481 \pm \sqrt{16081.99725651 - 4 \cdot 4.9382716 \cdot 2523.5}}{2 \cdot 4.9382716}$

Step 5.5.1.2

Multiply $- 4 \cdot 4.9382716 \cdot 2523.5$ .

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Step 5.5.1.2.1

Multiply $- 4$ by $4.9382716$ .

$b = \frac{126.81481481 \pm \sqrt{16081.99725651 - 19.75308641 \cdot 2523.5}}{2 \cdot 4.9382716}$

Step 5.5.1.2.2

Multiply $- 19.75308641$ by $2523.5$ .

$b = \frac{126.81481481 \pm \sqrt{16081.99725651 - 49846.91358024}}{2 \cdot 4.9382716}$

Step 5.5.1.3

Subtract $49846.91358024$ from $16081.99725651$ .

$b = \frac{126.81481481 \pm \sqrt{- 33764.91632373}}{2 \cdot 4.9382716}$

Step 5.5.1.4

Rewrite $- 33764.91632373$ as $- 1 (33764.91632373)$ .

$b = \frac{126.81481481 \pm \sqrt{- 1 \cdot 33764.91632373}}{2 \cdot 4.9382716}$

Step 5.5.1.5

Rewrite $\sqrt{- 1 (33764.91632373)}$ as $\sqrt{- 1} \cdot \sqrt{33764.91632373}$ .

$b = \frac{126.81481481 \pm \sqrt{- 1} \cdot \sqrt{33764.91632373}}{2 \cdot 4.9382716}$

Step 5.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$b = \frac{126.81481481 \pm i \sqrt{33764.91632373}}{2 \cdot 4.9382716}$

Step 5.5.2

Multiply $2$ by $4.9382716$ .

$b = \frac{126.81481481 \pm i \sqrt{33764.91632373}}{9.8765432}$

Step 5.5.3

Multiply by $1$ .

$b = \frac{1 (126.81481481 \pm i \sqrt{33764.91632373})}{9.8765432}$

Step 5.5.4

Factor $9.8765432$ out of $9.8765432$ .

$b = \frac{1 (126.81481481 \pm i \sqrt{33764.91632373})}{9.8765432 (1)}$

Step 5.5.5

Separate fractions.

$b = \frac{1}{9.8765432} \cdot \frac{126.81481481 \pm i \sqrt{33764.91632373}}{1}$

Step 5.5.6

Divide $1$ by $9.8765432$ .

$b = 0.10125 (\frac{126.81481481 \pm i \sqrt{33764.91632373}}{1})$

Step 5.5.7

Divide $126.81481481 \pm i \sqrt{33764.91632373}$ by $1$ .

$b = 0.10125 (126.81481481 \pm i \sqrt{33764.91632373})$

Step 5.6

The final answer is the combination of both solutions.

$b = 12.84 + 18.60492273 i, 12.84 - 18.60492273 i$