Algebra Examples

$\sqrt{2 x} + 2 = \sqrt{5 x - 4}$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{2 x} = \sqrt{5 x - 4} - 2$

Step 2

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

${\sqrt{2 x}}^{2} = {(\sqrt{5 x - 4} - 2)}^{2}$

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

${({(2 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{5 x - 4} - 2)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5 x - 4} - 2)}^{2}$

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.

${(2 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5 x - 4} - 2)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x)}^{1} = {(\sqrt{5 x - 4} - 2)}^{2}$

Step 3.2.1.2

Simplify.

$2 x = {(\sqrt{5 x - 4} - 2)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\sqrt{5 x - 4} - 2)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(\sqrt{5 x - 4} - 2)}^{2}$ as $(\sqrt{5 x - 4} - 2) (\sqrt{5 x - 4} - 2)$ .

$2 x = (\sqrt{5 x - 4} - 2) (\sqrt{5 x - 4} - 2)$

Step 3.3.1.2

Expand $(\sqrt{5 x - 4} - 2) (\sqrt{5 x - 4} - 2)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$2 x = \sqrt{5 x - 4} (\sqrt{5 x - 4} - 2) - 2 (\sqrt{5 x - 4} - 2)$

Step 3.3.1.2.2

Apply the distributive property.

$2 x = \sqrt{5 x - 4} \sqrt{5 x - 4} + \sqrt{5 x - 4} \cdot - 2 - 2 (\sqrt{5 x - 4} - 2)$

Step 3.3.1.2.3

Apply the distributive property.

$2 x = \sqrt{5 x - 4} \sqrt{5 x - 4} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

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 $\sqrt{5 x - 4} \sqrt{5 x - 4}$ .

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Step 3.3.1.3.1.1.1

Raise $\sqrt{5 x - 4}$ to the power of $1$ .

$2 x = {\sqrt{5 x - 4}}^{1} \sqrt{5 x - 4} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.1.2

Raise $\sqrt{5 x - 4}$ to the power of $1$ .

$2 x = {\sqrt{5 x - 4}}^{1} {\sqrt{5 x - 4}}^{1} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.1.3

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

$2 x = {\sqrt{5 x - 4}}^{1 + 1} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

$2 x = {\sqrt{5 x - 4}}^{2} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.2

Rewrite ${\sqrt{5 x - 4}}^{2}$ as $5 x - 4$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 4}$ as ${(5 x - 4)}^{\frac{1}{2}}$ .

$2 x = {({(5 x - 4)}^{\frac{1}{2}})}^{2} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.2.2

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

$2 x = {(5 x - 4)}^{\frac{1}{2} \cdot 2} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.2.3

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

$2 x = {(5 x - 4)}^{\frac{2}{2}} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

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.

$2 x = {(5 x - 4)}^{\frac{2}{2}} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

$2 x = {(5 x - 4)}^{1} + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.2.5

Simplify.

$2 x = 5 x - 4 + \sqrt{5 x - 4} \cdot - 2 - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.3

Move $- 2$ to the left of $\sqrt{5 x - 4}$ .

$2 x = 5 x - 4 - 2 \cdot \sqrt{5 x - 4} - 2 \sqrt{5 x - 4} - 2 \cdot - 2$

Step 3.3.1.3.1.4

Multiply $- 2$ by $- 2$ .

$2 x = 5 x - 4 - 2 \sqrt{5 x - 4} - 2 \sqrt{5 x - 4} + 4$

Step 3.3.1.3.2

Add $- 4$ and $4$ .

$2 x = 5 x + 0 - 2 \sqrt{5 x - 4} - 2 \sqrt{5 x - 4}$

Step 3.3.1.3.3

Add $5 x$ and $0$ .

$2 x = 5 x - 2 \sqrt{5 x - 4} - 2 \sqrt{5 x - 4}$

Step 3.3.1.3.4

Subtract $2 \sqrt{5 x - 4}$ from $- 2 \sqrt{5 x - 4}$ .

$2 x = 5 x - 4 \sqrt{5 x - 4}$

Step 4

Solve for $- 4 \sqrt{5 x - 4}$ .

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Step 4.1

Rewrite the equation as $5 x - 4 \sqrt{5 x - 4} = 2 x$ .

$5 x - 4 \sqrt{5 x - 4} = 2 x$

Step 4.2

Move all terms not containing $- 4 \sqrt{5 x - 4}$ to the right side of the equation.

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Step 4.2.1

Subtract $5 x$ from both sides of the equation.

$- 4 \sqrt{5 x - 4} = 2 x - 5 x$

Step 4.2.2

Subtract $5 x$ from $2 x$ .

$- 4 \sqrt{5 x - 4} = - 3 x$

Step 5

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

${(- 4 \sqrt{5 x - 4})}^{2} = {(- 3 x)}^{2}$

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 $\sqrt{5 x - 4}$ as ${(5 x - 4)}^{\frac{1}{2}}$ .

${(- 4 {(5 x - 4)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 4 {(5 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 4 {(5 x - 4)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(5 x - 4)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Step 6.2.1.2

Raise $- 4$ to the power of $2$ .

$16 {({(5 x - 4)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(5 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$16 {(5 x - 4)}^{\frac{1}{2} \cdot 2} = {(- 3 x)}^{2}$

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.

$16 {(5 x - 4)}^{\frac{1}{2} \cdot 2} = {(- 3 x)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(5 x - 4)}^{1} = {(- 3 x)}^{2}$

Step 6.2.1.4

Simplify.

$16 (5 x - 4) = {(- 3 x)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 (5 x) + 16 \cdot - 4 = {(- 3 x)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $5$ by $16$ .

$80 x + 16 \cdot - 4 = {(- 3 x)}^{2}$

Step 6.2.1.6.2

Multiply $16$ by $- 4$ .

$80 x - 64 = {(- 3 x)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

Apply the product rule to $- 3 x$ .

$80 x - 64 = {(- 3)}^{2} x^{2}$

Step 6.3.1.2

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

$80 x - 64 = 9 x^{2}$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $9 x^{2}$ from both sides of the equation.

$80 x - 64 - 9 x^{2} = 0$

Step 7.2

Factor the left side of the equation.

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Step 7.2.1

Factor $- 1$ out of $80 x - 64 - 9 x^{2}$ .

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Step 7.2.1.1

Reorder the expression.

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Step 7.2.1.1.1

Move $- 64$ .

$80 x - 9 x^{2} - 64 = 0$

Step 7.2.1.1.2

Reorder $80 x$ and $- 9 x^{2}$ .

$- 9 x^{2} + 80 x - 64 = 0$

Step 7.2.1.2

Factor $- 1$ out of $- 9 x^{2}$ .

$- (9 x^{2}) + 80 x - 64 = 0$

Step 7.2.1.3

Factor $- 1$ out of $80 x$ .

$- (9 x^{2}) - (- 80 x) - 64 = 0$

Step 7.2.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$- (9 x^{2}) - (- 80 x) - 1 \cdot 64 = 0$

Step 7.2.1.5

Factor $- 1$ out of $- (9 x^{2}) - (- 80 x)$ .

$- (9 x^{2} - 80 x) - 1 \cdot 64 = 0$

Step 7.2.1.6

Factor $- 1$ out of $- (9 x^{2} - 80 x) - 1 (64)$ .

$- (9 x^{2} - 80 x + 64) = 0$

Step 7.2.2

Factor.

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Step 7.2.2.1

Factor by grouping.

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Step 7.2.2.1.1

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 $a \cdot c = 9 \cdot 64 = 576$ and whose sum is $b = - 80$ .

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Step 7.2.2.1.1.1

Factor $- 80$ out of $- 80 x$ .

$- (9 x^{2} - 80 x + 64) = 0$

Step 7.2.2.1.1.2

Rewrite $- 80$ as $- 8$ plus $- 72$

$- (9 x^{2} + (- 8 - 72) x + 64) = 0$

Step 7.2.2.1.1.3

Apply the distributive property.

$- (9 x^{2} - 8 x - 72 x + 64) = 0$

Step 7.2.2.1.2

Factor out the greatest common factor from each group.

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Step 7.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((9 x^{2} - 8 x) - 72 x + 64) = 0$

Step 7.2.2.1.2.2

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

$- (x (9 x - 8) - 8 (9 x - 8)) = 0$

Step 7.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $9 x - 8$ .

$- ((9 x - 8) (x - 8)) = 0$

Step 7.2.2.2

Remove unnecessary parentheses.

$- (9 x - 8) (x - 8) = 0$

Step 7.3

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

$9 x - 8 = 0$

$x - 8 = 0$

Step 7.4

Set $9 x - 8$ equal to $0$ and solve for $x$ .

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Step 7.4.1

Set $9 x - 8$ equal to $0$ .

$9 x - 8 = 0$

Step 7.4.2

Solve $9 x - 8 = 0$ for $x$ .

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Step 7.4.2.1

Add $8$ to both sides of the equation.

$9 x = 8$

Step 7.4.2.2

Divide each term in $9 x = 8$ by $9$ and simplify.

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Step 7.4.2.2.1

Divide each term in $9 x = 8$ by $9$ .

$\frac{9 x}{9} = \frac{8}{9}$

Step 7.4.2.2.2

Simplify the left side.

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Step 7.4.2.2.2.1

Cancel the common factor of $9$ .

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Step 7.4.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{8}{9}$

Step 7.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{9}$

Step 7.5

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 7.5.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 7.5.2

Add $8$ to both sides of the equation.

$x = 8$

Step 7.6

The final solution is all the values that make $- (9 x - 8) (x - 8) = 0$ true.

$x = \frac{8}{9}, 8$

Step 8

Exclude the solutions that do not make $\sqrt{2 x} + 2 = \sqrt{5 x - 4}$ true.

$x = 8$