Basic Math Examples

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

Step 1

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

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

Step 2

Factor each term.

Tap for more steps...

Step 2.1

Combine exponents.

Tap for more steps...

Step 2.1.1

Multiply $2$ by $3.14$ .

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

Step 2.1.2

Multiply $0.6$ by $2$ .

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

Step 2.1.3

Multiply $0.6$ by $2$ .

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

Step 2.2

Simplify the denominator.

Tap for more steps...

Step 2.2.1

Factor $0.001$ out of $0.467 + 1.2 \cdot h$ .

Tap for more steps...

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.

Tap for more steps...

Step 2.2.4.1

Factor $0.03$ out of $0.33 + 1.2 h$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $(467 + 1200 h) (11 + 40 h)$ .

Tap for more steps...

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.

Tap for more steps...

Step 4.3.1

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

Tap for more steps...

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.

Tap for more steps...

Step 4.3.2.1

Simplify each term.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 5.5.1

Simplify the numerator.

Tap for more steps...

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

Tap for more steps...

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:

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