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.

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