Finite Math Examples

$L W = 48$ , $L^{2} + W^{2} = 100$

Step 1

Divide each term in $L W = 48$ by $W$ and simplify.

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Step 1.1

Divide each term in $L W = 48$ by $W$ .

$\frac{L W}{W} = \frac{48}{W}$

$L^{2} + W^{2} = 100$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $W$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{L W}{W} = \frac{48}{W}$

$L^{2} + W^{2} = 100$

Step 1.2.1.2

Divide $L$ by $1$ .

$L = \frac{48}{W}$

$L^{2} + W^{2} = 100$

$L = \frac{48}{W}$

$L^{2} + W^{2} = 100$

$L = \frac{48}{W}$

$L^{2} + W^{2} = 100$

$L = \frac{48}{W}$

$L^{2} + W^{2} = 100$

Step 2

Replace all occurrences of $L$ with $\frac{48}{W}$ in each equation.

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Step 2.1

Replace all occurrences of $L$ in $L^{2} + W^{2} = 100$ with $\frac{48}{W}$ .

${(\frac{48}{W})}^{2} + W^{2} = 100$

$L = \frac{48}{W}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the product rule to $\frac{48}{W}$ .

$\frac{48^{2}}{W^{2}} + W^{2} = 100$

$L = \frac{48}{W}$

Step 2.2.1.2

Raise $48$ to the power of $2$ .

$\frac{2304}{W^{2}} + W^{2} = 100$

$L = \frac{48}{W}$

$\frac{2304}{W^{2}} + W^{2} = 100$

$L = \frac{48}{W}$

$\frac{2304}{W^{2}} + W^{2} = 100$

$L = \frac{48}{W}$

$\frac{2304}{W^{2}} + W^{2} = 100$

$L = \frac{48}{W}$

Step 3

Solve for $W$ in $\frac{2304}{W^{2}} + W^{2} = 100$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

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

$W^{2}, 1, 1$

$L = \frac{48}{W}$

Step 3.1.2

The LCM of one and any expression is the expression.

$W^{2}$

$L = \frac{48}{W}$

$W^{2}$

$L = \frac{48}{W}$

Step 3.2

Multiply each term in $\frac{2304}{W^{2}} + W^{2} = 100$ by $W^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{2304}{W^{2}} + W^{2} = 100$ by $W^{2}$ .

$\frac{2304}{W^{2}} \cdot W^{2} + W^{2} W^{2} = 100 W^{2}$

$L = \frac{48}{W}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $W^{2}$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{2304}{W^{2}} \cdot W^{2} + W^{2} W^{2} = 100 W^{2}$

$L = \frac{48}{W}$

Step 3.2.2.1.1.2

Rewrite the expression.

$2304 + W^{2} W^{2} = 100 W^{2}$

$L = \frac{48}{W}$

$2304 + W^{2} W^{2} = 100 W^{2}$

$L = \frac{48}{W}$

Step 3.2.2.1.2

Multiply $W^{2}$ by $W^{2}$ by adding the exponents.

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Step 3.2.2.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2304 + W^{2 + 2} = 100 W^{2}$

$L = \frac{48}{W}$

Step 3.2.2.1.2.2

Add $2$ and $2$ .

$2304 + W^{4} = 100 W^{2}$

$L = \frac{48}{W}$

$2304 + W^{4} = 100 W^{2}$

$L = \frac{48}{W}$

$2304 + W^{4} = 100 W^{2}$

$L = \frac{48}{W}$

$2304 + W^{4} = 100 W^{2}$

$L = \frac{48}{W}$

$2304 + W^{4} = 100 W^{2}$

$L = \frac{48}{W}$

Step 3.3

Solve the equation.

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Step 3.3.1

Subtract $100 W^{2}$ from both sides of the equation.

$2304 + W^{4} - 100 W^{2} = 0$

$L = \frac{48}{W}$

Step 3.3.2

Substitute $u = W^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 100 u + 2304 = 0$

$u = W^{2}$

$L = \frac{48}{W}$

Step 3.3.3

Factor $u^{2} - 100 u + 2304$ using the AC method.

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Step 3.3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2304$ and whose sum is $- 100$ .

$- 64, - 36$

$L = \frac{48}{W}$

Step 3.3.3.2

Write the factored form using these integers.

$(u - 64) (u - 36) = 0$

$L = \frac{48}{W}$

$(u - 64) (u - 36) = 0$

$L = \frac{48}{W}$

Step 3.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 64 = 0$

$u - 36 = 0$

$L = \frac{48}{W}$

Step 3.3.5

Set $u - 64$ equal to $0$ and solve for $u$ .

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Step 3.3.5.1

Set $u - 64$ equal to $0$ .

$u - 64 = 0$

$L = \frac{48}{W}$

Step 3.3.5.2

Add $64$ to both sides of the equation.

$u = 64$

$L = \frac{48}{W}$

$u = 64$

$L = \frac{48}{W}$

Step 3.3.6

Set $u - 36$ equal to $0$ and solve for $u$ .

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Step 3.3.6.1

Set $u - 36$ equal to $0$ .

$u - 36 = 0$

$L = \frac{48}{W}$

Step 3.3.6.2

Add $36$ to both sides of the equation.

$u = 36$

$L = \frac{48}{W}$

$u = 36$

$L = \frac{48}{W}$

Step 3.3.7

The final solution is all the values that make $(u - 64) (u - 36) = 0$ true.

$u = 64, 36$

$L = \frac{48}{W}$

Step 3.3.8

Substitute the real value of $u = W^{2}$ back into the solved equation.

$W^{2} = 64$

$W^{2} = 36$

$L = \frac{48}{W}$

Step 3.3.9

Solve the first equation for $W$ .

$W^{2} = 64$

$L = \frac{48}{W}$

Step 3.3.10

Solve the equation for $W$ .

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Step 3.3.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$W = \pm \sqrt{64}$

$L = \frac{48}{W}$

Step 3.3.10.2

Simplify $\pm \sqrt{64}$ .

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Step 3.3.10.2.1

Rewrite $64$ as $8^{2}$ .

$W = \pm \sqrt{8^{2}}$

$L = \frac{48}{W}$

Step 3.3.10.2.2

Pull terms out from under the radical, assuming positive real numbers.

$W = \pm 8$

$L = \frac{48}{W}$

$W = \pm 8$

$L = \frac{48}{W}$

Step 3.3.10.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.10.3.1

First, use the positive value of the $\pm$ to find the first solution.

$W = 8$

$L = \frac{48}{W}$

Step 3.3.10.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$W = - 8$

$L = \frac{48}{W}$

Step 3.3.10.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$W = 8, - 8$

$L = \frac{48}{W}$

$W = 8, - 8$

$L = \frac{48}{W}$

$W = 8, - 8$

$L = \frac{48}{W}$

Step 3.3.11

Solve the second equation for $W$ .

$W^{2} = 36$

$L = \frac{48}{W}$

Step 3.3.12

Solve the equation for $W$ .

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Step 3.3.12.1

Remove parentheses.

$W^{2} = 36$

$L = \frac{48}{W}$

Step 3.3.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$W = \pm \sqrt{36}$

$L = \frac{48}{W}$

Step 3.3.12.3

Simplify $\pm \sqrt{36}$ .

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Step 3.3.12.3.1

Rewrite $36$ as $6^{2}$ .

$W = \pm \sqrt{6^{2}}$

$L = \frac{48}{W}$

Step 3.3.12.3.2

Pull terms out from under the radical, assuming positive real numbers.

$W = \pm 6$

$L = \frac{48}{W}$

$W = \pm 6$

$L = \frac{48}{W}$

Step 3.3.12.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.12.4.1

First, use the positive value of the $\pm$ to find the first solution.

$W = 6$

$L = \frac{48}{W}$

Step 3.3.12.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$W = - 6$

$L = \frac{48}{W}$

Step 3.3.12.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$W = 6, - 6$

$L = \frac{48}{W}$

$W = 6, - 6$

$L = \frac{48}{W}$

$W = 6, - 6$

$L = \frac{48}{W}$

Step 3.3.13

The solution to $W^{4} - 100 W^{2} + 2304 = 0$ is $W = 8, - 8, 6, - 6$ .

$W = 8, - 8, 6, - 6$

$L = \frac{48}{W}$

$W = 8, - 8, 6, - 6$

$L = \frac{48}{W}$

$W = 8, - 8, 6, - 6$

$L = \frac{48}{W}$

Step 4

Replace all occurrences of $W$ with $8$ in each equation.

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Step 4.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $8$ .

$L = \frac{48}{8}$

$W = 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $48$ by $8$ .

$L = 6$

$W = 8$

$L = 6$

$W = 8$

$L = 6$

$W = 8$

Step 5

Replace all occurrences of $W$ with $- 8$ in each equation.

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Step 5.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $- 8$ .

$L = \frac{48}{- 8}$

$W = - 8$

Step 5.2

Simplify the right side.

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Step 5.2.1

Divide $48$ by $- 8$ .

$L = - 6$

$W = - 8$

$L = - 6$

$W = - 8$

$L = - 6$

$W = - 8$

Step 6

Replace all occurrences of $W$ with $8$ in each equation.

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Step 6.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $8$ .

$L = \frac{48}{8}$

$W = 8$

Step 6.2

Simplify the right side.

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Step 6.2.1

Divide $48$ by $8$ .

$L = 6$

$W = 8$

$L = 6$

$W = 8$

$L = 6$

$W = 8$

Step 7

Replace all occurrences of $W$ with $- 8$ in each equation.

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Step 7.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $- 8$ .

$L = \frac{48}{- 8}$

$W = - 8$

Step 7.2

Simplify the right side.

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Step 7.2.1

Divide $48$ by $- 8$ .

$L = - 6$

$W = - 8$

$L = - 6$

$W = - 8$

$L = - 6$

$W = - 8$

Step 8

Replace all occurrences of $W$ with $6$ in each equation.

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Step 8.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $6$ .

$L = \frac{48}{6}$

$W = 6$

Step 8.2

Simplify the right side.

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Step 8.2.1

Divide $48$ by $6$ .

$L = 8$

$W = 6$

$L = 8$

$W = 6$

$L = 8$

$W = 6$

Step 9

Replace all occurrences of $W$ with $8$ in each equation.

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Step 9.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $8$ .

$L = \frac{48}{8}$

$W = 8$

Step 9.2

Simplify the right side.

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Step 9.2.1

Divide $48$ by $8$ .

$L = 6$

$W = 8$

$L = 6$

$W = 8$

$L = 6$

$W = 8$

Step 10

Replace all occurrences of $W$ with $- 8$ in each equation.

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Step 10.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $- 8$ .

$L = \frac{48}{- 8}$

$W = - 8$

Step 10.2

Simplify the right side.

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Step 10.2.1

Divide $48$ by $- 8$ .

$L = - 6$

$W = - 8$

$L = - 6$

$W = - 8$

$L = - 6$

$W = - 8$

Step 11

Replace all occurrences of $W$ with $6$ in each equation.

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Step 11.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $6$ .

$L = \frac{48}{6}$

$W = 6$

Step 11.2

Simplify the right side.

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Step 11.2.1

Divide $48$ by $6$ .

$L = 8$

$W = 6$

$L = 8$

$W = 6$

$L = 8$

$W = 6$

Step 12

Replace all occurrences of $W$ with $- 6$ in each equation.

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Step 12.1

Replace all occurrences of $W$ in $L = \frac{48}{W}$ with $- 6$ .

$L = \frac{48}{- 6}$

$W = - 6$

Step 12.2

Simplify the right side.

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Step 12.2.1

Divide $48$ by $- 6$ .

$L = - 8$

$W = - 6$

$L = - 8$

$W = - 6$

$L = - 8$

$W = - 6$

Step 13

List all of the solutions.

$L = 6, W = 8$

$L = - 6, W = - 8$

$L = 8, W = 6$

$L = - 8, W = - 6$