Finite Math Examples

$\sqrt{2 x} - 5 = \sqrt{3 x - 8}$

Step 1

Add $5$ to both sides of the equation.

$\sqrt{2 x} = \sqrt{3 x - 8} + 5$

Step 2

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

${\sqrt{2 x}}^{2} = {(\sqrt{3 x - 8} + 5)}^{2}$

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

${({(2 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{3 x - 8} + 5)}^{2}$

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

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

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Step 3.2.1.1.1

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

${(2 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{3 x - 8} + 5)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(2 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{3 x - 8} + 5)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x)}^{1} = {(\sqrt{3 x - 8} + 5)}^{2}$

Step 3.2.1.2

Simplify.

$2 x = {(\sqrt{3 x - 8} + 5)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\sqrt{3 x - 8} + 5)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(\sqrt{3 x - 8} + 5)}^{2}$ as $(\sqrt{3 x - 8} + 5) (\sqrt{3 x - 8} + 5)$ .

$2 x = (\sqrt{3 x - 8} + 5) (\sqrt{3 x - 8} + 5)$

Step 3.3.1.2

Expand $(\sqrt{3 x - 8} + 5) (\sqrt{3 x - 8} + 5)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$2 x = \sqrt{3 x - 8} (\sqrt{3 x - 8} + 5) + 5 (\sqrt{3 x - 8} + 5)$

Step 3.3.1.2.2

Apply the distributive property.

$2 x = \sqrt{3 x - 8} \sqrt{3 x - 8} + \sqrt{3 x - 8} \cdot 5 + 5 (\sqrt{3 x - 8} + 5)$

Step 3.3.1.2.3

Apply the distributive property.

$2 x = \sqrt{3 x - 8} \sqrt{3 x - 8} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $\sqrt{3 x - 8} \sqrt{3 x - 8}$ .

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Step 3.3.1.3.1.1.1

Raise $\sqrt{3 x - 8}$ to the power of $1$ .

$2 x = {\sqrt{3 x - 8}}^{1} \sqrt{3 x - 8} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.1.2

Raise $\sqrt{3 x - 8}$ to the power of $1$ .

$2 x = {\sqrt{3 x - 8}}^{1} {\sqrt{3 x - 8}}^{1} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.1.3

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

$2 x = {\sqrt{3 x - 8}}^{1 + 1} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

$2 x = {\sqrt{3 x - 8}}^{2} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.2

Rewrite ${\sqrt{3 x - 8}}^{2}$ as $3 x - 8$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 8}$ as ${(3 x - 8)}^{\frac{1}{2}}$ .

$2 x = {({(3 x - 8)}^{\frac{1}{2}})}^{2} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.2.2

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

$2 x = {(3 x - 8)}^{\frac{1}{2} \cdot 2} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.2.3

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

$2 x = {(3 x - 8)}^{\frac{2}{2}} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.4.1

Cancel the common factor.

$2 x = {(3 x - 8)}^{\frac{2}{2}} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

$2 x = {(3 x - 8)}^{1} + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.2.5

Simplify.

$2 x = 3 x - 8 + \sqrt{3 x - 8} \cdot 5 + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.3

Move $5$ to the left of $\sqrt{3 x - 8}$ .

$2 x = 3 x - 8 + 5 \cdot \sqrt{3 x - 8} + 5 \sqrt{3 x - 8} + 5 \cdot 5$

Step 3.3.1.3.1.4

Multiply $5$ by $5$ .

$2 x = 3 x - 8 + 5 \sqrt{3 x - 8} + 5 \sqrt{3 x - 8} + 25$

Step 3.3.1.3.2

Add $- 8$ and $25$ .

$2 x = 3 x + 17 + 5 \sqrt{3 x - 8} + 5 \sqrt{3 x - 8}$

Step 3.3.1.3.3

Add $5 \sqrt{3 x - 8}$ and $5 \sqrt{3 x - 8}$ .

$2 x = 3 x + 17 + 10 \sqrt{3 x - 8}$

Step 4

Solve for $10 \sqrt{3 x - 8}$ .

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Step 4.1

Rewrite the equation as $3 x + 17 + 10 \sqrt{3 x - 8} = 2 x$ .

$3 x + 17 + 10 \sqrt{3 x - 8} = 2 x$

Step 4.2

Move all terms not containing $10 \sqrt{3 x - 8}$ to the right side of the equation.

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Step 4.2.1

Subtract $3 x$ from both sides of the equation.

$17 + 10 \sqrt{3 x - 8} = 2 x - 3 x$

Step 4.2.2

Subtract $17$ from both sides of the equation.

$10 \sqrt{3 x - 8} = 2 x - 3 x - 17$

Step 4.2.3

Subtract $3 x$ from $2 x$ .

$10 \sqrt{3 x - 8} = - x - 17$

Step 5

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

${(10 \sqrt{3 x - 8})}^{2} = {(- x - 17)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 8}$ as ${(3 x - 8)}^{\frac{1}{2}}$ .

${(10 {(3 x - 8)}^{\frac{1}{2}})}^{2} = {(- x - 17)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(10 {(3 x - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $10 {(3 x - 8)}^{\frac{1}{2}}$ .

$10^{2} {({(3 x - 8)}^{\frac{1}{2}})}^{2} = {(- x - 17)}^{2}$

Step 6.2.1.2

Raise $10$ to the power of $2$ .

$100 {({(3 x - 8)}^{\frac{1}{2}})}^{2} = {(- x - 17)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(3 x - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$100 {(3 x - 8)}^{\frac{1}{2} \cdot 2} = {(- x - 17)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$100 {(3 x - 8)}^{\frac{1}{2} \cdot 2} = {(- x - 17)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$100 {(3 x - 8)}^{1} = {(- x - 17)}^{2}$

Step 6.2.1.4

Simplify.

$100 (3 x - 8) = {(- x - 17)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$100 (3 x) + 100 \cdot - 8 = {(- x - 17)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $3$ by $100$ .

$300 x + 100 \cdot - 8 = {(- x - 17)}^{2}$

Step 6.2.1.6.2

Multiply $100$ by $- 8$ .

$300 x - 800 = {(- x - 17)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(- x - 17)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(- x - 17)}^{2}$ as $(- x - 17) (- x - 17)$ .

$300 x - 800 = (- x - 17) (- x - 17)$

Step 6.3.1.2

Expand $(- x - 17) (- x - 17)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$300 x - 800 = - x (- x - 17) - 17 (- x - 17)$

Step 6.3.1.2.2

Apply the distributive property.

$300 x - 800 = - x (- x) - x \cdot - 17 - 17 (- x - 17)$

Step 6.3.1.2.3

Apply the distributive property.

$300 x - 800 = - x (- x) - x \cdot - 17 - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$300 x - 800 = - 1 \cdot - 1 x \cdot x - x \cdot - 17 - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$300 x - 800 = - 1 \cdot - 1 (x \cdot x) - x \cdot - 17 - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$300 x - 800 = - 1 \cdot - 1 x^{2} - x \cdot - 17 - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$300 x - 800 = 1 x^{2} - x \cdot - 17 - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3.1.4

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

$300 x - 800 = x^{2} - x \cdot - 17 - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3.1.5

Multiply $- 17$ by $- 1$ .

$300 x - 800 = x^{2} + 17 x - 17 (- x) - 17 \cdot - 17$

Step 6.3.1.3.1.6

Multiply $- 1$ by $- 17$ .

$300 x - 800 = x^{2} + 17 x + 17 x - 17 \cdot - 17$

Step 6.3.1.3.1.7

Multiply $- 17$ by $- 17$ .

$300 x - 800 = x^{2} + 17 x + 17 x + 289$

Step 6.3.1.3.2

Add $17 x$ and $17 x$ .

$300 x - 800 = x^{2} + 34 x + 289$

Step 7

Solve for $x$ .

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Step 7.1

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

$x^{2} + 34 x + 289 = 300 x - 800$

Step 7.2

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

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Step 7.2.1

Subtract $300 x$ from both sides of the equation.

$x^{2} + 34 x + 289 - 300 x = - 800$

Step 7.2.2

Subtract $300 x$ from $34 x$ .

$x^{2} - 266 x + 289 = - 800$

Step 7.3

Move all terms to the left side of the equation and simplify.

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Step 7.3.1

Add $800$ to both sides of the equation.

$x^{2} - 266 x + 289 + 800 = 0$

Step 7.3.2

Add $289$ and $800$ .

$x^{2} - 266 x + 1089 = 0$

Step 7.4

Use the quadratic formula to find the solutions.

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

Step 7.5

Substitute the values $a = 1$ , $b = - 266$ , and $c = 1089$ into the quadratic formula and solve for $x$ .

$\frac{266 \pm \sqrt{{(- 266)}^{2} - 4 \cdot (1 \cdot 1089)}}{2 \cdot 1}$

Step 7.6

Simplify.

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Step 7.6.1

Simplify the numerator.

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Step 7.6.1.1

Raise $- 266$ to the power of $2$ .

$x = \frac{266 \pm \sqrt{70756 - 4 \cdot 1 \cdot 1089}}{2 \cdot 1}$

Step 7.6.1.2

Multiply $- 4 \cdot 1 \cdot 1089$ .

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Step 7.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{266 \pm \sqrt{70756 - 4 \cdot 1089}}{2 \cdot 1}$

Step 7.6.1.2.2

Multiply $- 4$ by $1089$ .

$x = \frac{266 \pm \sqrt{70756 - 4356}}{2 \cdot 1}$

Step 7.6.1.3

Subtract $4356$ from $70756$ .

$x = \frac{266 \pm \sqrt{66400}}{2 \cdot 1}$

Step 7.6.1.4

Rewrite $66400$ as $20^{2} \cdot 166$ .

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Step 7.6.1.4.1

Factor $400$ out of $66400$ .

$x = \frac{266 \pm \sqrt{400 (166)}}{2 \cdot 1}$

Step 7.6.1.4.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{266 \pm \sqrt{20^{2} \cdot 166}}{2 \cdot 1}$

Step 7.6.1.5

Pull terms out from under the radical.

$x = \frac{266 \pm 20 \sqrt{166}}{2 \cdot 1}$

Step 7.6.2

Multiply $2$ by $1$ .

$x = \frac{266 \pm 20 \sqrt{166}}{2}$

Step 7.6.3

Simplify $\frac{266 \pm 20 \sqrt{166}}{2}$ .

$x = 133 \pm 10 \sqrt{166}$

Step 7.7

The final answer is the combination of both solutions.

$x = 133 + 10 \sqrt{166}, 133 - 10 \sqrt{166}$

Step 8

Exclude the solutions that do not make $\sqrt{2 x} - 5 = \sqrt{3 x - 8}$ true.

$No solution$