Algebra Examples

$\sqrt{x} + 1 = \sqrt{9 x - 2}$

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{x} = \sqrt{9 x - 2} - 1$

Step 2

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

${\sqrt{x}}^{2} = {(\sqrt{9 x - 2} - 1)}^{2}$

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\sqrt{9 x - 2} - 1)}^{2}$

Step 3.2.1.2

Simplify.

$x = {(\sqrt{9 x - 2} - 1)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\sqrt{9 x - 2} - 1)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(\sqrt{9 x - 2} - 1)}^{2}$ as $(\sqrt{9 x - 2} - 1) (\sqrt{9 x - 2} - 1)$ .

$x = (\sqrt{9 x - 2} - 1) (\sqrt{9 x - 2} - 1)$

Step 3.3.1.2

Expand $(\sqrt{9 x - 2} - 1) (\sqrt{9 x - 2} - 1)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x = \sqrt{9 x - 2} (\sqrt{9 x - 2} - 1) - 1 (\sqrt{9 x - 2} - 1)$

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

$x = \sqrt{9 x - 2} \sqrt{9 x - 2} + \sqrt{9 x - 2} \cdot - 1 - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

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 $\sqrt{9 x - 2} \sqrt{9 x - 2}$ .

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Step 3.3.1.3.1.1.1

Raise $\sqrt{9 x - 2}$ to the power of $1$ .

$x = {\sqrt{9 x - 2}}^{1} \sqrt{9 x - 2} + \sqrt{9 x - 2} \cdot - 1 - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.1.2

Raise $\sqrt{9 x - 2}$ to the power of $1$ .

$x = {\sqrt{9 x - 2}}^{1} {\sqrt{9 x - 2}}^{1} + \sqrt{9 x - 2} \cdot - 1 - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.1.3

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

$x = {\sqrt{9 x - 2}}^{1 + 1} + \sqrt{9 x - 2} \cdot - 1 - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

$x = {\sqrt{9 x - 2}}^{2} + \sqrt{9 x - 2} \cdot - 1 - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.2

Rewrite ${\sqrt{9 x - 2}}^{2}$ as $9 x - 2$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x - 2}$ as ${(9 x - 2)}^{\frac{1}{2}}$ .

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

Step 3.3.1.3.1.2.2

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

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

Step 3.3.1.3.1.2.3

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

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

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.

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

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.2.5

Simplify.

$x = 9 x - 2 + \sqrt{9 x - 2} \cdot - 1 - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.3

Move $- 1$ to the left of $\sqrt{9 x - 2}$ .

$x = 9 x - 2 - 1 \cdot \sqrt{9 x - 2} - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.4

Rewrite $- 1 \sqrt{9 x - 2}$ as $- \sqrt{9 x - 2}$ .

$x = 9 x - 2 - \sqrt{9 x - 2} - 1 \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.5

Rewrite $- 1 \sqrt{9 x - 2}$ as $- \sqrt{9 x - 2}$ .

$x = 9 x - 2 - \sqrt{9 x - 2} - \sqrt{9 x - 2} - 1 \cdot - 1$

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$x = 9 x - 2 - \sqrt{9 x - 2} - \sqrt{9 x - 2} + 1$

Step 3.3.1.3.2

Add $- 2$ and $1$ .

$x = 9 x - 1 - \sqrt{9 x - 2} - \sqrt{9 x - 2}$

Step 3.3.1.3.3

Subtract $\sqrt{9 x - 2}$ from $- \sqrt{9 x - 2}$ .

$x = 9 x - 1 - 2 \sqrt{9 x - 2}$

Step 4

Solve for $- 2 \sqrt{9 x - 2}$ .

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Step 4.1

Rewrite the equation as $9 x - 1 - 2 \sqrt{9 x - 2} = x$ .

$9 x - 1 - 2 \sqrt{9 x - 2} = x$

Step 4.2

Move all terms not containing $- 2 \sqrt{9 x - 2}$ to the right side of the equation.

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Step 4.2.1

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

$- 1 - 2 \sqrt{9 x - 2} = x - 9 x$

Step 4.2.2

Add $1$ to both sides of the equation.

$- 2 \sqrt{9 x - 2} = x - 9 x + 1$

Step 4.2.3

Subtract $9 x$ from $x$ .

$- 2 \sqrt{9 x - 2} = - 8 x + 1$

Step 5

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

${(- 2 \sqrt{9 x - 2})}^{2} = {(- 8 x + 1)}^{2}$

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 $\sqrt{9 x - 2}$ as ${(9 x - 2)}^{\frac{1}{2}}$ .

${(- 2 {(9 x - 2)}^{\frac{1}{2}})}^{2} = {(- 8 x + 1)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 2 {(9 x - 2)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 2 {(9 x - 2)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(9 x - 2)}^{\frac{1}{2}})}^{2} = {(- 8 x + 1)}^{2}$

Step 6.2.1.2

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

$4 {({(9 x - 2)}^{\frac{1}{2}})}^{2} = {(- 8 x + 1)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(9 x - 2)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

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

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.

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

Step 6.2.1.3.2.2

Rewrite the expression.

$4 {(9 x - 2)}^{1} = {(- 8 x + 1)}^{2}$

Step 6.2.1.4

Simplify.

$4 (9 x - 2) = {(- 8 x + 1)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$4 (9 x) + 4 \cdot - 2 = {(- 8 x + 1)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $9$ by $4$ .

$36 x + 4 \cdot - 2 = {(- 8 x + 1)}^{2}$

Step 6.2.1.6.2

Multiply $4$ by $- 2$ .

$36 x - 8 = {(- 8 x + 1)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(- 8 x + 1)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(- 8 x + 1)}^{2}$ as $(- 8 x + 1) (- 8 x + 1)$ .

$36 x - 8 = (- 8 x + 1) (- 8 x + 1)$

Step 6.3.1.2

Expand $(- 8 x + 1) (- 8 x + 1)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$36 x - 8 = - 8 x (- 8 x + 1) + 1 (- 8 x + 1)$

Step 6.3.1.2.2

Apply the distributive property.

$36 x - 8 = - 8 x (- 8 x) - 8 x \cdot 1 + 1 (- 8 x + 1)$

Step 6.3.1.2.3

Apply the distributive property.

$36 x - 8 = - 8 x (- 8 x) - 8 x \cdot 1 + 1 (- 8 x) + 1 \cdot 1$

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.

$36 x - 8 = - 8 \cdot - 8 x \cdot x - 8 x \cdot 1 + 1 (- 8 x) + 1 \cdot 1$

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 $x$ .

$36 x - 8 = - 8 \cdot - 8 (x \cdot x) - 8 x \cdot 1 + 1 (- 8 x) + 1 \cdot 1$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$36 x - 8 = - 8 \cdot - 8 x^{2} - 8 x \cdot 1 + 1 (- 8 x) + 1 \cdot 1$

Step 6.3.1.3.1.3

Multiply $- 8$ by $- 8$ .

$36 x - 8 = 64 x^{2} - 8 x \cdot 1 + 1 (- 8 x) + 1 \cdot 1$

Step 6.3.1.3.1.4

Multiply $- 8$ by $1$ .

$36 x - 8 = 64 x^{2} - 8 x + 1 (- 8 x) + 1 \cdot 1$

Step 6.3.1.3.1.5

Multiply $- 8 x$ by $1$ .

$36 x - 8 = 64 x^{2} - 8 x - 8 x + 1 \cdot 1$

Step 6.3.1.3.1.6

Multiply $1$ by $1$ .

$36 x - 8 = 64 x^{2} - 8 x - 8 x + 1$

Step 6.3.1.3.2

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

$36 x - 8 = 64 x^{2} - 16 x + 1$

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.

$64 x^{2} - 16 x + 1 = 36 x - 8$

Step 7.2

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

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Step 7.2.1

Subtract $36 x$ from both sides of the equation.

$64 x^{2} - 16 x + 1 - 36 x = - 8$

Step 7.2.2

Subtract $36 x$ from $- 16 x$ .

$64 x^{2} - 52 x + 1 = - 8$

Step 7.3

Add $8$ to both sides of the equation.

$64 x^{2} - 52 x + 1 + 8 = 0$

Step 7.4

Add $1$ and $8$ .

$64 x^{2} - 52 x + 9 = 0$

Step 7.5

Factor by grouping.

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Step 7.5.1

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 $a \cdot c = 64 \cdot 9 = 576$ and whose sum is $b = - 52$ .

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Step 7.5.1.1

Factor $- 52$ out of $- 52 x$ .

$64 x^{2} - 52 x + 9 = 0$

Step 7.5.1.2

Rewrite $- 52$ as $- 16$ plus $- 36$

$64 x^{2} + (- 16 - 36) x + 9 = 0$

Step 7.5.1.3

Apply the distributive property.

$64 x^{2} - 16 x - 36 x + 9 = 0$

Step 7.5.2

Factor out the greatest common factor from each group.

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Step 7.5.2.1

Group the first two terms and the last two terms.

$(64 x^{2} - 16 x) - 36 x + 9 = 0$

Step 7.5.2.2

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

$16 x (4 x - 1) - 9 (4 x - 1) = 0$

Step 7.5.3

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

$(4 x - 1) (16 x - 9) = 0$

Step 7.6

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

$4 x - 1 = 0$

$16 x - 9 = 0$

Step 7.7

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

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Step 7.7.1

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

$4 x - 1 = 0$

Step 7.7.2

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

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Step 7.7.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 7.7.2.2

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

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Step 7.7.2.2.1

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

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

Step 7.7.2.2.2

Simplify the left side.

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Step 7.7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.7.2.2.2.1.1

Cancel the common factor.

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

Step 7.7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 7.8

Set $16 x - 9$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $16 x - 9$ equal to $0$ .

$16 x - 9 = 0$

Step 7.8.2

Solve $16 x - 9 = 0$ for $x$ .

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Step 7.8.2.1

Add $9$ to both sides of the equation.

$16 x = 9$

Step 7.8.2.2

Divide each term in $16 x = 9$ by $16$ and simplify.

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Step 7.8.2.2.1

Divide each term in $16 x = 9$ by $16$ .

$\frac{16 x}{16} = \frac{9}{16}$

Step 7.8.2.2.2

Simplify the left side.

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Step 7.8.2.2.2.1

Cancel the common factor of $16$ .

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Step 7.8.2.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{9}{16}$

Step 7.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{16}$

Step 7.9

The final solution is all the values that make $(4 x - 1) (16 x - 9) = 0$ true.

$x = \frac{1}{4}, \frac{9}{16}$

Step 8

Exclude the solutions that do not make $\sqrt{x} + 1 = \sqrt{9 x - 2}$ true.

$x = \frac{9}{16}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{16}$

Decimal Form:

$x = 0.5625$