Finite Math Examples

$\frac{5000000}{6000} = {(1 + x)}^{2}$

Step 1

Rewrite the equation as ${(1 + x)}^{2} = \frac{5000000}{6000}$ .

${(1 + x)}^{2} = \frac{5000000}{6000}$

Step 2

Reduce the expression $\frac{5000000}{6000}$ by cancelling the common factors.

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Step 2.1

Factor $2000$ out of $5000000$ .

${(1 + x)}^{2} = \frac{2000 (2500)}{6000}$

Step 2.2

Factor $2000$ out of $6000$ .

${(1 + x)}^{2} = \frac{2000 \cdot 2500}{2000 \cdot 3}$

Step 2.3

Cancel the common factor.

${(1 + x)}^{2} = \frac{2000 \cdot 2500}{2000 \cdot 3}$

Step 2.4

Rewrite the expression.

${(1 + x)}^{2} = \frac{2500}{3}$

Step 3

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

$1 + x = \pm \sqrt{\frac{2500}{3}}$

Step 4

Simplify $\pm \sqrt{\frac{2500}{3}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{2500}{3}}$ as $\frac{\sqrt{2500}}{\sqrt{3}}$ .

$1 + x = \pm \frac{\sqrt{2500}}{\sqrt{3}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $2500$ as $50^{2}$ .

$1 + x = \pm \frac{\sqrt{50^{2}}}{\sqrt{3}}$

Step 4.2.2

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

$1 + x = \pm \frac{50}{\sqrt{3}}$

Step 4.3

Multiply $\frac{50}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$1 + x = \pm \frac{50}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{50}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$1 + x = \pm \frac{50 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$1 + x = \pm \frac{50 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$1 + x = \pm \frac{50 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.4.4

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

$1 + x = \pm \frac{50 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$1 + x = \pm \frac{50 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$1 + x = \pm \frac{50 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 + x = \pm \frac{50 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$1 + x = \pm \frac{50 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$1 + x = \pm \frac{50 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$1 + x = \pm \frac{50 \sqrt{3}}{3^{1}}$

Step 4.4.6.5

Evaluate the exponent.

$1 + x = \pm \frac{50 \sqrt{3}}{3}$

Step 5

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

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Step 5.1

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

$1 + x = \frac{50 \sqrt{3}}{3}$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = \frac{50 \sqrt{3}}{3} - 1$

Step 5.3

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

$1 + x = - \frac{50 \sqrt{3}}{3}$

Step 5.4

Subtract $1$ from both sides of the equation.

$x = - \frac{50 \sqrt{3}}{3} - 1$

Step 5.5

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

$x = \frac{50 \sqrt{3}}{3} - 1, - \frac{50 \sqrt{3}}{3} - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{50 \sqrt{3}}{3} - 1, - \frac{50 \sqrt{3}}{3} - 1$

Decimal Form:

$x = 27.86751345 \dots, - 29.86751345 \dots$