Basic Math Examples

23 = \frac{120}{2 \cdot 3.14 \cdot ((0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6))}

$23 = \frac{120}{2 \cdot 3.14 \cdot ((0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6))}$

Step 1

Rewrite the equation as

\frac{120}{2 \cdot 3.14 \cdot ((0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6))} = 23

$\frac{120}{2 \cdot 3.14 \cdot ((0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6))} = 23$ .

\frac{120}{2 \cdot 3.14 \cdot ((0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6))} = 23

$\frac{120}{2 \cdot 3.14 \cdot ((0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6))} = 23$

Step 2

Factor each term.

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Step 2.1

Combine exponents.

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Step 2.1.1

Multiply

2

$2$ by

3.14

$3.14$ .

\frac{120}{6.28 \cdot (0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6)} = 23

$\frac{120}{6.28 \cdot (0.467 + 2 \cdot h \cdot 0.6) (0.33 + 2 \cdot h \cdot 0.6)} = 23$

Step 2.1.2

Multiply

0.6

$0.6$ by

2

$2$ .

\frac{120}{6.28 \cdot (0.467 + 1.2 \cdot h) (0.33 + 2 \cdot h \cdot 0.6)} = 23

$\frac{120}{6.28 \cdot (0.467 + 1.2 \cdot h) (0.33 + 2 \cdot h \cdot 0.6)} = 23$

Step 2.1.3

Multiply

0.6

$0.6$ by

2

$2$ .

\frac{120}{6.28 \cdot (0.467 + 1.2 \cdot h) (0.33 + 1.2 \cdot h)} = 23

$\frac{120}{6.28 \cdot (0.467 + 1.2 \cdot h) (0.33 + 1.2 \cdot h)} = 23$

\frac{120}{6.28 \cdot (0.467 + 1.2 \cdot h) (0.33 + 1.2 \cdot h)} = 23

$\frac{120}{6.28 \cdot (0.467 + 1.2 \cdot h) (0.33 + 1.2 \cdot h)} = 23$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Factor

$0.001$ out of

$0.467 + 1.2 \cdot h$ .

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Step 2.2.1.1

Factor

$0.001$ out of

$0.467$ .

$\frac{120}{6.28 \cdot (0.001 (467) + 1.2 \cdot h) (0.33 + 1.2 \cdot h)} = 23$

Step 2.2.1.2

Factor

$0.001$ out of

$1.2 \cdot h$ .

$\frac{120}{6.28 \cdot (0.001 (467) + 0.001 (1200 \cdot h)) (0.33 + 1.2 \cdot h)} = 23$

Step 2.2.1.3

Factor

$0.001$ out of

$0.001 (467) + 0.001 (1200 \cdot h)$ .

$\frac{120}{6.28 \cdot (0.001 (467 + 1200 \cdot h)) (0.33 + 1.2 \cdot h)} = 23$

$\frac{120}{6.28 \cdot (0.001 (467 + 1200 h)) (0.33 + 1.2 \cdot h)} = 23$

Step 2.2.2

Remove unnecessary parentheses.

$\frac{120}{6.28 \cdot 0.001 (467 + 1200 h) (0.33 + 1.2 \cdot h)} = 23$

Step 2.2.3

Multiply

$6.28$ by

$0.001$ .

$\frac{120}{0.00628 (467 + 1200 h) (0.33 + 1.2 h)} = 23$

Step 2.2.4

Factor.

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Step 2.2.4.1

Factor

$0.03$ out of

$0.33 + 1.2 h$ .

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Step 2.2.4.1.1

Factor

$0.03$ out of

$0.33$ .

$\frac{120}{0.00628 (467 + 1200 h) (0.03 (11) + 1.2 h)} = 23$

Step 2.2.4.1.2

Factor

$0.03$ out of

$1.2 h$ .

$\frac{120}{0.00628 (467 + 1200 h) (0.03 (11) + 0.03 (40 h))} = 23$

Step 2.2.4.1.3

Factor

$0.03$ out of

$0.03 (11) + 0.03 (40 h)$ .

$\frac{120}{0.00628 (467 + 1200 h) (0.03 (11 + 40 h))} = 23$

Step 2.2.4.2

Remove unnecessary parentheses.

$\frac{120}{0.00628 (467 + 1200 h) \cdot 0.03 (11 + 40 h)} = 23$

Step 2.2.5

Multiply

$0.03$ by

$0.00628$ .

$\frac{120}{0.0001884 (467 + 1200 h) (11 + 40 h)} = 23$

Step 2.3

Factor

$120$ out of

$120$ .

$\frac{120 (1)}{0.0001884 (467 + 1200 h) (11 + 40 h)} = 23$

Step 2.4

Factor

$0.0001884$ out of

$0.0001884 (467 + 1200 h) (11 + 40 h)$ .

$\frac{120 (1)}{0.0001884 ((467 + 1200 h) (11 + 40 h))} = 23$

Step 2.5

Separate fractions.

$\frac{120}{0.0001884} \cdot \frac{1}{(467 + 1200 h) (11 + 40 h)} = 23$

Step 2.6

Divide

$120$ by

$0.0001884$ .

$636942.67515923 \frac{1}{(467 + 1200 h) (11 + 40 h)} = 23$

Step 2.7

Combine

$636942.67515923$ and

$\frac{1}{(467 + 1200 h) (11 + 40 h)}$ .

$\frac{636942.67515923}{(467 + 1200 h) (11 + 40 h)} = 23$

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.

$(467 + 1200 h) (11 + 40 h), 1$

Step 3.2

The LCM of one and any expression is the expression.

$(467 + 1200 h) (11 + 40 h)$

Step 4

Multiply each term in

$\frac{636942.67515923}{(467 + 1200 h) (11 + 40 h)} = 23$ by

$(467 + 1200 h) (11 + 40 h)$ to eliminate the fractions.

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Step 4.1

Multiply each term in

$\frac{636942.67515923}{(467 + 1200 h) (11 + 40 h)} = 23$ by

$(467 + 1200 h) (11 + 40 h)$ .

$\frac{636942.67515923}{(467 + 1200 h) (11 + 40 h)} ((467 + 1200 h) (11 + 40 h)) = 23 ((467 + 1200 h) (11 + 40 h))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

$(467 + 1200 h) (11 + 40 h)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{636942.67515923}{(467 + 1200 h) (11 + 40 h)} ((467 + 1200 h) (11 + 40 h)) = 23 ((467 + 1200 h) (11 + 40 h))$

Step 4.2.1.2

Rewrite the expression.

$636942.67515923 = 23 ((467 + 1200 h) (11 + 40 h))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Expand

$(467 + 1200 h) (11 + 40 h)$ using the FOIL Method.

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Step 4.3.1.1

Apply the distributive property.

$636942.67515923 = 23 (467 (11 + 40 h) + 1200 h (11 + 40 h))$

Step 4.3.1.2

Apply the distributive property.

$636942.67515923 = 23 (467 \cdot 11 + 467 (40 h) + 1200 h (11 + 40 h))$

Step 4.3.1.3

Apply the distributive property.

$636942.67515923 = 23 (467 \cdot 11 + 467 (40 h) + 1200 h \cdot 11 + 1200 h (40 h))$

Step 4.3.2

Simplify and combine like terms.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply

$467$ by

$11$ .

$636942.67515923 = 23 (5137 + 467 (40 h) + 1200 h \cdot 11 + 1200 h (40 h))$

Step 4.3.2.1.2

Multiply

$40$ by

$467$ .

$636942.67515923 = 23 (5137 + 18680 h + 1200 h \cdot 11 + 1200 h (40 h))$

Step 4.3.2.1.3

Multiply

$11$ by

$1200$ .

$636942.67515923 = 23 (5137 + 18680 h + 13200 h + 1200 h (40 h))$

Step 4.3.2.1.4

Rewrite using the commutative property of multiplication.

$636942.67515923 = 23 (5137 + 18680 h + 13200 h + 1200 \cdot 40 h \cdot h)$

Step 4.3.2.1.5

Multiply

$h$ by

$h$ by adding the exponents.

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Step 4.3.2.1.5.1

Move

$h$ .

$636942.67515923 = 23 (5137 + 18680 h + 13200 h + 1200 \cdot 40 (h \cdot h))$

Step 4.3.2.1.5.2

Multiply

$h$ by

$h$ .

$636942.67515923 = 23 (5137 + 18680 h + 13200 h + 1200 \cdot 40 h^{2})$

Step 4.3.2.1.6

Multiply

$1200$ by

$40$ .

$636942.67515923 = 23 (5137 + 18680 h + 13200 h + 48000 h^{2})$

Step 4.3.2.2

Add

$18680 h$ and

$13200 h$ .

$636942.67515923 = 23 (5137 + 31880 h + 48000 h^{2})$

Step 4.3.3

Apply the distributive property.

$636942.67515923 = 23 \cdot 5137 + 23 (31880 h) + 23 (48000 h^{2})$

Step 4.3.4

Simplify.

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Step 4.3.4.1

Multiply

$23$ by

$5137$ .

$636942.67515923 = 118151 + 23 (31880 h) + 23 (48000 h^{2})$

Step 4.3.4.2

Multiply

$31880$ by

$23$ .

$636942.67515923 = 118151 + 733240 h + 23 (48000 h^{2})$

Step 4.3.4.3

Multiply

$48000$ by

$23$ .

$636942.67515923 = 118151 + 733240 h + 1104000 h^{2}$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as

$118151 + 733240 h + 1104000 h^{2} = 636942.67515923$ .

$118151 + 733240 h + 1104000 h^{2} = 636942.67515923$

Step 5.2

Move all terms to the left side of the equation and simplify.

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Step 5.2.1

Subtract

$636942.67515923$ from both sides of the equation.

$118151 + 733240 h + 1104000 h^{2} - 636942.67515923 = 0$

Step 5.2.2

Subtract

$636942.67515923$ from

$118151$ .

$733240 h + 1104000 h^{2} - 518791.67515923 = 0$

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

$a = 1104000$ ,

$b = 733240$ , and

$c = - 518791.67515923$ into the quadratic formula and solve for

$h$ .

$\frac{- 733240 \pm \sqrt{733240^{2} - 4 \cdot (1104000 \cdot - 518791.67515923)}}{2 \cdot 1104000}$

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

$733240$ to the power of

$2$ .

$h = \frac{- 733240 \pm \sqrt{537640897600 - 4 \cdot 1104000 \cdot - 518791.67515923}}{2 \cdot 1104000}$

Step 5.5.1.2

Multiply

$- 4 \cdot 1104000 \cdot - 518791.67515923$ .

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Step 5.5.1.2.1

Multiply

$- 4$ by

$1104000$ .

$h = \frac{- 733240 \pm \sqrt{537640897600 - 4416000 \cdot - 518791.67515923}}{2 \cdot 1104000}$

Step 5.5.1.2.2

Multiply

$- 4416000$ by

$- 518791.67515923$ .

$h = \frac{- 733240 \pm \sqrt{537640897600 + 2290984037503.18471337}}{2 \cdot 1104000}$

Step 5.5.1.3

Add

$537640897600$ and

$2290984037503.18471337$ .

$h = \frac{- 733240 \pm \sqrt{2828624935103.18471337}}{2 \cdot 1104000}$

Step 5.5.2

Multiply

$2$ by

$1104000$ .

$h = \frac{- 733240 \pm \sqrt{2828624935103.18471337}}{2208000}$

Step 5.6

The final answer is the combination of both solutions.

$h = - \frac{733240 - \sqrt{2828624935103.18471337}}{2208000}, - \frac{733240 + \sqrt{2828624935103.18471337}}{2208000}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$h = - \frac{733240 - \sqrt{2828624935103.18471337}}{2208000}, - \frac{733240 + \sqrt{2828624935103.18471337}}{2208000}$

Decimal Form:

$h = 0.42962483 \dots, - 1.09379150 \dots$