Finite Math Examples

$25 x^{- 4} - 104 x^{- 2} + 16 = 0$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$25 \frac{1}{x^{4}} - 104 x^{- 2} + 16 = 0$

Step 1.2

Combine $25$ and $\frac{1}{x^{4}}$ .

$\frac{25}{x^{4}} - 104 x^{- 2} + 16 = 0$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{25}{x^{4}} - 104 \frac{1}{x^{2}} + 16 = 0$

Step 1.4

Combine $- 104$ and $\frac{1}{x^{2}}$ .

$\frac{25}{x^{4}} + \frac{- 104}{x^{2}} + 16 = 0$

Step 1.5

Move the negative in front of the fraction.

$\frac{25}{x^{4}} - \frac{104}{x^{2}} + 16 = 0$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$x^{4}, x^{2}, 1, 1$

Step 2.2

Since $x^{4}, x^{2}, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{4}, x^{2}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factors for $x^{4}$ are $x \cdot x \cdot x \cdot x$ , which is $x$ multiplied by each other $4$ times.

$x^{4} = x \cdot x \cdot x \cdot x$

$x$ occurs $4$ times.

Step 2.7

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 2.8

The LCM of $x^{4}, x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x \cdot x$

Step 2.9

Simplify $x \cdot x \cdot x \cdot x$ .

Tap for more steps...

Step 2.9.1

Multiply $x$ by $x$ .

$x^{2} \cdot x \cdot x$

Step 2.9.2

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

Tap for more steps...

Step 2.9.2.1

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 2.9.2.1.1

Raise $x$ to the power of $1$ .

$x^{2} \cdot x^{1} \cdot x$

Step 2.9.2.1.2

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

$x^{2 + 1} \cdot x$

Step 2.9.2.2

Add $2$ and $1$ .

$x^{3} \cdot x$

Step 2.9.3

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.9.3.1

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 2.9.3.1.1

Raise $x$ to the power of $1$ .

$x^{3} \cdot x^{1}$

Step 2.9.3.1.2

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

$x^{3 + 1}$

Step 2.9.3.2

Add $3$ and $1$ .

$x^{4}$

Step 3

Multiply each term in $\frac{25}{x^{4}} - \frac{104}{x^{2}} + 16 = 0$ by $x^{4}$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{25}{x^{4}} - \frac{104}{x^{2}} + 16 = 0$ by $x^{4}$ .

$\frac{25}{x^{4}} x^{4} - \frac{104}{x^{2}} x^{4} + 16 x^{4} = 0 x^{4}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of $x^{4}$ .

Tap for more steps...

Step 3.2.1.1.1

Cancel the common factor.

$\frac{25}{x^{4}} x^{4} - \frac{104}{x^{2}} x^{4} + 16 x^{4} = 0 x^{4}$

Step 3.2.1.1.2

Rewrite the expression.

$25 - \frac{104}{x^{2}} x^{4} + 16 x^{4} = 0 x^{4}$

Step 3.2.1.2

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

Tap for more steps...

Step 3.2.1.2.1

Move the leading negative in $- \frac{104}{x^{2}}$ into the numerator.

$25 + \frac{- 104}{x^{2}} x^{4} + 16 x^{4} = 0 x^{4}$

Step 3.2.1.2.2

Factor $x^{2}$ out of $x^{4}$ .

$25 + \frac{- 104}{x^{2}} (x^{2} x^{2}) + 16 x^{4} = 0 x^{4}$

Step 3.2.1.2.3

Cancel the common factor.

$25 + \frac{- 104}{x^{2}} (x^{2} x^{2}) + 16 x^{4} = 0 x^{4}$

Step 3.2.1.2.4

Rewrite the expression.

$25 - 104 x^{2} + 16 x^{4} = 0 x^{4}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $0$ by $x^{4}$ .

$25 - 104 x^{2} + 16 x^{4} = 0$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

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

$16 u^{2} - 104 u + 25 = 0$

$u = x^{2}$

Step 4.2

Factor by grouping.

Tap for more steps...

Step 4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 16 \cdot 25 = 400$ and whose sum is $b = - 104$ .

Tap for more steps...

Step 4.2.1.1

Factor $- 104$ out of $- 104 u$ .

$16 u^{2} - 104 u + 25 = 0$

Step 4.2.1.2

Rewrite $- 104$ as $- 4$ plus $- 100$

$16 u^{2} + (- 4 - 100) u + 25 = 0$

Step 4.2.1.3

Apply the distributive property.

$16 u^{2} - 4 u - 100 u + 25 = 0$

Step 4.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.2.1

Group the first two terms and the last two terms.

$(16 u^{2} - 4 u) - 100 u + 25 = 0$

Step 4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$4 u (4 u - 1) - 25 (4 u - 1) = 0$

Step 4.2.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (4 u - 25) = 0$

Step 4.3

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

$4 u - 1 = 0$

$4 u - 25 = 0$

Step 4.4

Set $4 u - 1$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 4.4.1

Set $4 u - 1$ equal to $0$ .

$4 u - 1 = 0$

Step 4.4.2

Solve $4 u - 1 = 0$ for $u$ .

Tap for more steps...

Step 4.4.2.1

Add $1$ to both sides of the equation.

$4 u = 1$

Step 4.4.2.2

Divide each term in $4 u = 1$ by $4$ and simplify.

Tap for more steps...

Step 4.4.2.2.1

Divide each term in $4 u = 1$ by $4$ .

$\frac{4 u}{4} = \frac{1}{4}$

Step 4.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.4.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{1}{4}$

Step 4.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{4}$

Step 4.5

Set $4 u - 25$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 4.5.1

Set $4 u - 25$ equal to $0$ .

$4 u - 25 = 0$

Step 4.5.2

Solve $4 u - 25 = 0$ for $u$ .

Tap for more steps...

Step 4.5.2.1

Add $25$ to both sides of the equation.

$4 u = 25$

Step 4.5.2.2

Divide each term in $4 u = 25$ by $4$ and simplify.

Tap for more steps...

Step 4.5.2.2.1

Divide each term in $4 u = 25$ by $4$ .

$\frac{4 u}{4} = \frac{25}{4}$

Step 4.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.5.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{25}{4}$

Step 4.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{25}{4}$

Step 4.6

The final solution is all the values that make $(4 u - 1) (4 u - 25) = 0$ true.

$u = \frac{1}{4}, \frac{25}{4}$

Step 4.7

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

$x^{2} = \frac{1}{4}$

${(x^{2})}^{1} = \frac{25}{4}$

Step 4.8

Solve the first equation for $x$ .

$x^{2} = \frac{1}{4}$

Step 4.9

Solve the equation for $x$ .

Tap for more steps...

Step 4.9.1

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

$x = \pm \sqrt{\frac{1}{4}}$

Step 4.9.2

Simplify $\pm \sqrt{\frac{1}{4}}$ .

Tap for more steps...

Step 4.9.2.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 4.9.2.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 4.9.2.3

Simplify the denominator.

Tap for more steps...

Step 4.9.2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.9.2.3.2

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

$x = \pm \frac{1}{2}$

Step 4.9.3

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

Tap for more steps...

Step 4.9.3.1

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

$x = \frac{1}{2}$

Step 4.9.3.2

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

$x = - \frac{1}{2}$

Step 4.9.3.3

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

$x = \frac{1}{2}, - \frac{1}{2}$

Step 4.10

Solve the second equation for $x$ .

${(x^{2})}^{1} = \frac{25}{4}$

Step 4.11

Solve the equation for $x$ .

Tap for more steps...

Step 4.11.1

Remove parentheses.

$x^{2} = \frac{25}{4}$

Step 4.11.2

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

$x = \pm \sqrt{\frac{25}{4}}$

Step 4.11.3

Simplify $\pm \sqrt{\frac{25}{4}}$ .

Tap for more steps...

Step 4.11.3.1

Rewrite $\sqrt{\frac{25}{4}}$ as $\frac{\sqrt{25}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{25}}{\sqrt{4}}$

Step 4.11.3.2

Simplify the numerator.

Tap for more steps...

Step 4.11.3.2.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{\sqrt{5^{2}}}{\sqrt{4}}$

Step 4.11.3.2.2

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

$x = \pm \frac{5}{\sqrt{4}}$

Step 4.11.3.3

Simplify the denominator.

Tap for more steps...

Step 4.11.3.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{5}{\sqrt{2^{2}}}$

Step 4.11.3.3.2

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

$x = \pm \frac{5}{2}$

Step 4.11.4

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

Tap for more steps...

Step 4.11.4.1

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

$x = \frac{5}{2}$

Step 4.11.4.2

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

$x = - \frac{5}{2}$

Step 4.11.4.3

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

$x = \frac{5}{2}, - \frac{5}{2}$

Step 4.12

The solution to $16 x^{4} - 104 x^{2} + 25 = 0$ is $x = \frac{1}{2}, - \frac{1}{2}, \frac{5}{2}, - \frac{5}{2}$ .

$x = \frac{1}{2}, - \frac{1}{2}, \frac{5}{2}, - \frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, - \frac{1}{2}, \frac{5}{2}, - \frac{5}{2}$

Decimal Form:

$x = 0.5, - 0.5, 2.5, - 2.5$

Mixed Number Form:

$x = \frac{1}{2}, - \frac{1}{2}, 2 \frac{1}{2}, - 2 \frac{1}{2}$