Calculus Examples

$\frac{x}{4} + 2 \sqrt{x} + 15 = x$

Step 1

Move all terms not containing $2 \sqrt{x}$ to the right side of the equation.

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Step 1.1

Subtract $\frac{x}{4}$ from both sides of the equation.

$2 \sqrt{x} + 15 = x - \frac{x}{4}$

Step 1.2

Subtract $15$ from both sides of the equation.

$2 \sqrt{x} = x - \frac{x}{4} - 15$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x})}^{2} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(2 x^{\frac{1}{2}})}^{2} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(2 x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $2 x^{\frac{1}{2}}$ .

$2^{2} {(x^{\frac{1}{2}})}^{2} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.2.1.2

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

$4 {(x^{\frac{1}{2}})}^{2} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

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

$4 x^{\frac{1}{2} \cdot 2} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$4 x^{1} = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.2.1.4

Simplify.

$4 x = {(x - \frac{x}{4} - 15)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(x - \frac{x}{4} - 15)}^{2}$ .

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Step 3.3.1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$4 x = {(x \cdot \frac{4}{4} - \frac{x}{4} - 15)}^{2}$

Step 3.3.1.2

Combine $x$ and $\frac{4}{4}$ .

$4 x = {(\frac{x \cdot 4}{4} - \frac{x}{4} - 15)}^{2}$

Step 3.3.1.3

Combine the numerators over the common denominator.

$4 x = {(\frac{x \cdot 4 - x}{4} - 15)}^{2}$

Step 3.3.1.4

Subtract $x$ from $x \cdot 4$ .

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Step 3.3.1.4.1

Reorder $x$ and $4$ .

$4 x = {(\frac{4 \cdot x - x}{4} - 15)}^{2}$

Step 3.3.1.4.2

Subtract $x$ from $4 \cdot x$ .

$4 x = {(\frac{3 \cdot x}{4} - 15)}^{2}$

Step 3.3.1.5

Rewrite ${(\frac{3 x}{4} - 15)}^{2}$ as $(\frac{3 x}{4} - 15) (\frac{3 x}{4} - 15)$ .

$4 x = (\frac{3 x}{4} - 15) (\frac{3 x}{4} - 15)$

Step 3.3.1.6

Expand $(\frac{3 x}{4} - 15) (\frac{3 x}{4} - 15)$ using the FOIL Method.

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Step 3.3.1.6.1

Apply the distributive property.

$4 x = \frac{3 x}{4} (\frac{3 x}{4} - 15) - 15 (\frac{3 x}{4} - 15)$

Step 3.3.1.6.2

Apply the distributive property.

$4 x = \frac{3 x}{4} \cdot \frac{3 x}{4} + \frac{3 x}{4} \cdot - 15 - 15 (\frac{3 x}{4} - 15)$

Step 3.3.1.6.3

Apply the distributive property.

$4 x = \frac{3 x}{4} \cdot \frac{3 x}{4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7

Simplify and combine like terms.

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Step 3.3.1.7.1

Simplify each term.

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Step 3.3.1.7.1.1

Multiply $\frac{3 x}{4} \cdot \frac{3 x}{4}$ .

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Step 3.3.1.7.1.1.1

Multiply $\frac{3 x}{4}$ by $\frac{3 x}{4}$ .

$4 x = \frac{3 x (3 x)}{4 \cdot 4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.1.2

Multiply $3$ by $3$ .

$4 x = \frac{9 x \cdot x}{4 \cdot 4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.1.3

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

$4 x = \frac{9 (x^{1} x)}{4 \cdot 4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.1.4

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

$4 x = \frac{9 (x^{1} x^{1})}{4 \cdot 4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.1.5

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

$4 x = \frac{9 x^{1 + 1}}{4 \cdot 4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.1.6

Add $1$ and $1$ .

$4 x = \frac{9 x^{2}}{4 \cdot 4} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.1.7

Multiply $4$ by $4$ .

$4 x = \frac{9 x^{2}}{16} + \frac{3 x}{4} \cdot - 15 - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.2

Multiply $\frac{3 x}{4} \cdot - 15$ .

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Step 3.3.1.7.1.2.1

Combine $\frac{3 x}{4}$ and $- 15$ .

$4 x = \frac{9 x^{2}}{16} + \frac{3 x \cdot - 15}{4} - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.2.2

Multiply $- 15$ by $3$ .

$4 x = \frac{9 x^{2}}{16} + \frac{- 45 x}{4} - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.3

Move the negative in front of the fraction.

$4 x = \frac{9 x^{2}}{16} - \frac{45 x}{4} - 15 \frac{3 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.4

Multiply $- 15 \frac{3 x}{4}$ .

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Step 3.3.1.7.1.4.1

Combine $- 15$ and $\frac{3 x}{4}$ .

$4 x = \frac{9 x^{2}}{16} - \frac{45 x}{4} + \frac{- 15 (3 x)}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.4.2

Multiply $3$ by $- 15$ .

$4 x = \frac{9 x^{2}}{16} - \frac{45 x}{4} + \frac{- 45 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.5

Move the negative in front of the fraction.

$4 x = \frac{9 x^{2}}{16} - \frac{45 x}{4} - \frac{45 x}{4} - 15 \cdot - 15$

Step 3.3.1.7.1.6

Multiply $- 15$ by $- 15$ .

$4 x = \frac{9 x^{2}}{16} - \frac{45 x}{4} - \frac{45 x}{4} + 225$

Step 3.3.1.7.2

Subtract $\frac{45 x}{4}$ from $- \frac{45 x}{4}$ .

$4 x = \frac{9 x^{2}}{16} - 2 \frac{45 x}{4} + 225$

Step 3.3.1.8

Simplify each term.

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Step 3.3.1.8.1

Cancel the common factor of $2$ .

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Step 3.3.1.8.1.1

Factor $2$ out of $- 2$ .

$4 x = \frac{9 x^{2}}{16} + 2 (- 1) \frac{45 x}{4} + 225$

Step 3.3.1.8.1.2

Factor $2$ out of $4$ .

$4 x = \frac{9 x^{2}}{16} + 2 \cdot - 1 \frac{45 x}{2 \cdot 2} + 225$

Step 3.3.1.8.1.3

Cancel the common factor.

$4 x = \frac{9 x^{2}}{16} + 2 \cdot - 1 \frac{45 x}{2 \cdot 2} + 225$

Step 3.3.1.8.1.4

Rewrite the expression.

$4 x = \frac{9 x^{2}}{16} - 1 \frac{45 x}{2} + 225$

Step 3.3.1.8.2

Rewrite $- 1 \frac{45 x}{2}$ as $- \frac{45 x}{2}$ .

$4 x = \frac{9 x^{2}}{16} - \frac{45 x}{2} + 225$

Step 4

Solve for $x$ .

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{9 x^{2}}{16} - \frac{45 x}{2} + 225 = 4 x$

Step 4.2

Move all terms containing $x$ to the left side of the equation.

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Step 4.2.1

Subtract $4 x$ from both sides of the equation.

$\frac{9 x^{2}}{16} - \frac{45 x}{2} + 225 - 4 x = 0$

Step 4.2.2

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9 x^{2}}{16} - \frac{45 x}{2} - 4 x \cdot \frac{2}{2} + 225 = 0$

Step 4.2.3

Combine $- 4 x$ and $\frac{2}{2}$ .

$\frac{9 x^{2}}{16} - \frac{45 x}{2} + \frac{- 4 x \cdot 2}{2} + 225 = 0$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{9 x^{2}}{16} + \frac{- 45 x - 4 x \cdot 2}{2} + 225 = 0$

Step 4.2.5

Find the common denominator.

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Step 4.2.5.1

Multiply $\frac{- 45 x - 4 x \cdot 2}{2}$ by $\frac{8}{8}$ .

$\frac{9 x^{2}}{16} + \frac{- 45 x - 4 x \cdot 2}{2} \cdot \frac{8}{8} + 225 = 0$

Step 4.2.5.2

Multiply $\frac{- 45 x - 4 x \cdot 2}{2}$ by $\frac{8}{8}$ .

$\frac{9 x^{2}}{16} + \frac{(- 45 x - 4 x \cdot 2) \cdot 8}{2 \cdot 8} + 225 = 0$

Step 4.2.5.3

Write $225$ as a fraction with denominator $1$ .

$\frac{9 x^{2}}{16} + \frac{(- 45 x - 4 x \cdot 2) \cdot 8}{2 \cdot 8} + \frac{225}{1} = 0$

Step 4.2.5.4

Multiply $\frac{225}{1}$ by $\frac{16}{16}$ .

$\frac{9 x^{2}}{16} + \frac{(- 45 x - 4 x \cdot 2) \cdot 8}{2 \cdot 8} + \frac{225}{1} \cdot \frac{16}{16} = 0$

Step 4.2.5.5

Multiply $\frac{225}{1}$ by $\frac{16}{16}$ .

$\frac{9 x^{2}}{16} + \frac{(- 45 x - 4 x \cdot 2) \cdot 8}{2 \cdot 8} + \frac{225 \cdot 16}{16} = 0$

Step 4.2.5.6

Multiply $2$ by $8$ .

$\frac{9 x^{2}}{16} + \frac{(- 45 x - 4 x \cdot 2) \cdot 8}{16} + \frac{225 \cdot 16}{16} = 0$

Step 4.2.6

Combine the numerators over the common denominator.

$\frac{9 x^{2} + (- 45 x - 4 x \cdot 2) \cdot 8 + 225 \cdot 16}{16} = 0$

Step 4.2.7

Simplify each term.

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Step 4.2.7.1

Multiply $2$ by $- 4$ .

$\frac{9 x^{2} + (- 45 x - 8 x) \cdot 8 + 225 \cdot 16}{16} = 0$

Step 4.2.7.2

Subtract $8 x$ from $- 45 x$ .

$\frac{9 x^{2} - 53 x \cdot 8 + 225 \cdot 16}{16} = 0$

Step 4.2.7.3

Multiply $8$ by $- 53$ .

$\frac{9 x^{2} - 424 x + 225 \cdot 16}{16} = 0$

Step 4.2.7.4

Multiply $225$ by $16$ .

$\frac{9 x^{2} - 424 x + 3600}{16} = 0$

Step 4.2.8

Factor by grouping.

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Step 4.2.8.1

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 $a \cdot c = 9 \cdot 3600 = 32400$ and whose sum is $b = - 424$ .

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Step 4.2.8.1.1

Factor $- 424$ out of $- 424 x$ .

$\frac{9 x^{2} - 424 (x) + 3600}{16} = 0$

Step 4.2.8.1.2

Rewrite $- 424$ as $- 100$ plus $- 324$

$\frac{9 x^{2} + (- 100 - 324) x + 3600}{16} = 0$

Step 4.2.8.1.3

Apply the distributive property.

$\frac{9 x^{2} - 100 x - 324 x + 3600}{16} = 0$

Step 4.2.8.2

Factor out the greatest common factor from each group.

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Step 4.2.8.2.1

Group the first two terms and the last two terms.

$\frac{(9 x^{2} - 100 x) - 324 x + 3600}{16} = 0$

Step 4.2.8.2.2

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

$\frac{x (9 x - 100) - 36 (9 x - 100)}{16} = 0$

Step 4.2.8.3

Factor the polynomial by factoring out the greatest common factor, $9 x - 100$ .

$\frac{(9 x - 100) (x - 36)}{16} = 0$

Step 4.3

Set the numerator equal to zero.

$(9 x - 100) (x - 36) = 0$

Step 4.4

Solve the equation for $x$ .

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Step 4.4.1

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

$9 x - 100 = 0$

$x - 36 = 0$

Step 4.4.2

Set $9 x - 100$ equal to $0$ and solve for $x$ .

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Step 4.4.2.1

Set $9 x - 100$ equal to $0$ .

$9 x - 100 = 0$

Step 4.4.2.2

Solve $9 x - 100 = 0$ for $x$ .

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Step 4.4.2.2.1

Add $100$ to both sides of the equation.

$9 x = 100$

Step 4.4.2.2.2

Divide each term in $9 x = 100$ by $9$ and simplify.

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Step 4.4.2.2.2.1

Divide each term in $9 x = 100$ by $9$ .

$\frac{9 x}{9} = \frac{100}{9}$

Step 4.4.2.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.2.1

Cancel the common factor of $9$ .

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Step 4.4.2.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{100}{9}$

Step 4.4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{100}{9}$

Step 4.4.3

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

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Step 4.4.3.1

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

$x - 36 = 0$

Step 4.4.3.2

Add $36$ to both sides of the equation.

$x = 36$

Step 4.4.4

The final solution is all the values that make $(9 x - 100) (x - 36) = 0$ true.

$x = \frac{100}{9}, 36$

Step 5

Exclude the solutions that do not make $\frac{x}{4} + 2 \sqrt{x} + 15 = x$ true.

$x = 36$