Finite Math Examples

$\sqrt{x^{2} - 10 x + 25} + 12 \sqrt{x} = 15 \sqrt{x}$

Step 1

Subtract $15 \sqrt{x}$ from both sides of the equation.

$\sqrt{x^{2} - 10 x + 25} + 12 \sqrt{x} - 15 \sqrt{x} = 0$

Step 2

Simplify $\sqrt{x^{2} - 10 x + 25} + 12 \sqrt{x} - 15 \sqrt{x}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor using the perfect square rule.

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Step 2.1.1.1

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

$\sqrt{x^{2} - 10 x + 5^{2}} + 12 \sqrt{x} - 15 \sqrt{x} = 0$

Step 2.1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 2.1.1.3

Rewrite the polynomial.

$\sqrt{x^{2} - 2 \cdot x \cdot 5 + 5^{2}} + 12 \sqrt{x} - 15 \sqrt{x} = 0$

Step 2.1.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$\sqrt{{(x - 5)}^{2}} + 12 \sqrt{x} - 15 \sqrt{x} = 0$

Step 2.1.2

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

$x - 5 + 12 \sqrt{x} - 15 \sqrt{x} = 0$

Step 2.2

Subtract $15 \sqrt{x}$ from $12 \sqrt{x}$ .

$x - 5 - 3 \sqrt{x} = 0$

Step 3

Move all terms not containing $- 3 \sqrt{x}$ to the right side of the equation.

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Step 3.1

Subtract $x$ from both sides of the equation.

$- 5 - 3 \sqrt{x} = - x$

Step 3.2

Add $5$ to both sides of the equation.

$- 3 \sqrt{x} = - x + 5$

Step 4

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

${(- 3 \sqrt{x})}^{2} = {(- x + 5)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(- 3 x^{\frac{1}{2}})}^{2} = {(- x + 5)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- 3 x^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $- 3 x^{\frac{1}{2}}$ .

${(- 3)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 5)}^{2}$

Step 5.2.1.2

Raise $- 3$ to the power of $2$ .

$9 {(x^{\frac{1}{2}})}^{2} = {(- x + 5)}^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$9 x^{\frac{1}{2} \cdot 2} = {(- x + 5)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$9 x^{\frac{1}{2} \cdot 2} = {(- x + 5)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$9 x^{1} = {(- x + 5)}^{2}$

Step 5.2.1.4

Simplify.

$9 x = {(- x + 5)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(- x + 5)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(- x + 5)}^{2}$ as $(- x + 5) (- x + 5)$ .

$9 x = (- x + 5) (- x + 5)$

Step 5.3.1.2

Expand $(- x + 5) (- x + 5)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$9 x = - x (- x + 5) + 5 (- x + 5)$

Step 5.3.1.2.2

Apply the distributive property.

$9 x = - x (- x) - x \cdot 5 + 5 (- x + 5)$

Step 5.3.1.2.3

Apply the distributive property.

$9 x = - x (- x) - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$9 x = - 1 \cdot - 1 x \cdot x - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.3.1.2.1

Move $x$ .

$9 x = - 1 \cdot - 1 (x \cdot x) - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$9 x = - 1 \cdot - 1 x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$9 x = 1 x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$9 x = x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3.1.5

Multiply $5$ by $- 1$ .

$9 x = x^{2} - 5 x + 5 (- x) + 5 \cdot 5$

Step 5.3.1.3.1.6

Multiply $- 1$ by $5$ .

$9 x = x^{2} - 5 x - 5 x + 5 \cdot 5$

Step 5.3.1.3.1.7

Multiply $5$ by $5$ .

$9 x = x^{2} - 5 x - 5 x + 25$

Step 5.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$9 x = x^{2} - 10 x + 25$

Step 6

Solve for $x$ .

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Step 6.1

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

$x^{2} - 10 x + 25 = 9 x$

Step 6.2

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

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Step 6.2.1

Subtract $9 x$ from both sides of the equation.

$x^{2} - 10 x + 25 - 9 x = 0$

Step 6.2.2

Subtract $9 x$ from $- 10 x$ .

$x^{2} - 19 x + 25 = 0$

Step 6.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.4

Substitute the values $a = 1$ , $b = - 19$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 6.5

Simplify.

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 6.5.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 6.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 25}}{2 \cdot 1}$

Step 6.5.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{19 \pm \sqrt{361 - 100}}{2 \cdot 1}$

Step 6.5.1.3

Subtract $100$ from $361$ .

$x = \frac{19 \pm \sqrt{261}}{2 \cdot 1}$

Step 6.5.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 6.5.1.4.1

Factor $9$ out of $261$ .

$x = \frac{19 \pm \sqrt{9 (29)}}{2 \cdot 1}$

Step 6.5.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{19 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot 1}$

Step 6.5.1.5

Pull terms out from under the radical.

$x = \frac{19 \pm 3 \sqrt{29}}{2 \cdot 1}$

Step 6.5.2

Multiply $2$ by $1$ .

$x = \frac{19 \pm 3 \sqrt{29}}{2}$

Step 6.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.6.1

Simplify the numerator.

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Step 6.6.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 6.6.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 6.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 25}}{2 \cdot 1}$

Step 6.6.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{19 \pm \sqrt{361 - 100}}{2 \cdot 1}$

Step 6.6.1.3

Subtract $100$ from $361$ .

$x = \frac{19 \pm \sqrt{261}}{2 \cdot 1}$

Step 6.6.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 6.6.1.4.1

Factor $9$ out of $261$ .

$x = \frac{19 \pm \sqrt{9 (29)}}{2 \cdot 1}$

Step 6.6.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{19 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot 1}$

Step 6.6.1.5

Pull terms out from under the radical.

$x = \frac{19 \pm 3 \sqrt{29}}{2 \cdot 1}$

Step 6.6.2

Multiply $2$ by $1$ .

$x = \frac{19 \pm 3 \sqrt{29}}{2}$

Step 6.6.3

Change the $\pm$ to $+$ .

$x = \frac{19 + 3 \sqrt{29}}{2}$

Step 6.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.7.1

Simplify the numerator.

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Step 6.7.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 6.7.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 6.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 25}}{2 \cdot 1}$

Step 6.7.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{19 \pm \sqrt{361 - 100}}{2 \cdot 1}$

Step 6.7.1.3

Subtract $100$ from $361$ .

$x = \frac{19 \pm \sqrt{261}}{2 \cdot 1}$

Step 6.7.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 6.7.1.4.1

Factor $9$ out of $261$ .

$x = \frac{19 \pm \sqrt{9 (29)}}{2 \cdot 1}$

Step 6.7.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{19 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot 1}$

Step 6.7.1.5

Pull terms out from under the radical.

$x = \frac{19 \pm 3 \sqrt{29}}{2 \cdot 1}$

Step 6.7.2

Multiply $2$ by $1$ .

$x = \frac{19 \pm 3 \sqrt{29}}{2}$

Step 6.7.3

Change the $\pm$ to $-$ .

$x = \frac{19 - 3 \sqrt{29}}{2}$

Step 6.8

The final answer is the combination of both solutions.

$x = \frac{19 + 3 \sqrt{29}}{2}, \frac{19 - 3 \sqrt{29}}{2}$

Step 7

Exclude the solutions that do not make $x - 5 - 3 \sqrt{x} = 0$ true.

$x = \frac{19 + 3 \sqrt{29}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19 + 3 \sqrt{29}}{2}$

Decimal Form:

$x = 17.57774721 \dots$