Algebra Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

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

Step 2.2.1.3.2.2

Rewrite the expression.

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

Step 2.2.1.4

Simplify.

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

Step 2.2.1.5

Apply the distributive property.

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

Step 2.2.1.6

Multiply $4$ by $- 4$ .

$4 x - 16 = {(- \sqrt{x - 1} + 4)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(- \sqrt{x - 1} + 4)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(- \sqrt{x - 1} + 4)}^{2}$ as $(- \sqrt{x - 1} + 4) (- \sqrt{x - 1} + 4)$ .

$4 x - 16 = (- \sqrt{x - 1} + 4) (- \sqrt{x - 1} + 4)$

Step 2.3.1.2

Expand $(- \sqrt{x - 1} + 4) (- \sqrt{x - 1} + 4)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$4 x - 16 = - \sqrt{x - 1} (- \sqrt{x - 1} + 4) + 4 (- \sqrt{x - 1} + 4)$

Step 2.3.1.2.2

Apply the distributive property.

$4 x - 16 = - \sqrt{x - 1} (- \sqrt{x - 1}) - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1} + 4)$

Step 2.3.1.2.3

Apply the distributive property.

$4 x - 16 = - \sqrt{x - 1} (- \sqrt{x - 1}) - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $- \sqrt{x - 1} (- \sqrt{x - 1})$ .

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Step 2.3.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$4 x - 16 = 1 \sqrt{x - 1} \sqrt{x - 1} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.1.2

Multiply $\sqrt{x - 1}$ by $1$ .

$4 x - 16 = \sqrt{x - 1} \sqrt{x - 1} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.1.3

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

$4 x - 16 = {\sqrt{x - 1}}^{1} \sqrt{x - 1} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.1.4

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

$4 x - 16 = {\sqrt{x - 1}}^{1} {\sqrt{x - 1}}^{1} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.1.5

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

$4 x - 16 = {\sqrt{x - 1}}^{1 + 1} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.1.6

Add $1$ and $1$ .

$4 x - 16 = {\sqrt{x - 1}}^{2} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

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

Step 2.3.1.3.1.2.2

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

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

Step 2.3.1.3.1.2.3

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

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

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$4 x - 16 = {(x - 1)}^{1} - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.2.5

Simplify.

$4 x - 16 = x - 1 - \sqrt{x - 1} \cdot 4 + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.3

Multiply $4$ by $- 1$ .

$4 x - 16 = x - 1 - 4 \sqrt{x - 1} + 4 (- \sqrt{x - 1}) + 4 \cdot 4$

Step 2.3.1.3.1.4

Multiply $- 1$ by $4$ .

$4 x - 16 = x - 1 - 4 \sqrt{x - 1} - 4 \sqrt{x - 1} + 4 \cdot 4$

Step 2.3.1.3.1.5

Multiply $4$ by $4$ .

$4 x - 16 = x - 1 - 4 \sqrt{x - 1} - 4 \sqrt{x - 1} + 16$

Step 2.3.1.3.2

Add $- 1$ and $16$ .

$4 x - 16 = x + 15 - 4 \sqrt{x - 1} - 4 \sqrt{x - 1}$

Step 2.3.1.3.3

Subtract $4 \sqrt{x - 1}$ from $- 4 \sqrt{x - 1}$ .

$4 x - 16 = x + 15 - 8 \sqrt{x - 1}$

Step 3

Solve for $- 8 \sqrt{x - 1}$ .

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Step 3.1

Rewrite the equation as $x + 15 - 8 \sqrt{x - 1} = 4 x - 16$ .

$x + 15 - 8 \sqrt{x - 1} = 4 x - 16$

Step 3.2

Move all terms not containing $- 8 \sqrt{x - 1}$ to the right side of the equation.

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Step 3.2.1

Subtract $x$ from both sides of the equation.

$15 - 8 \sqrt{x - 1} = 4 x - 16 - x$

Step 3.2.2

Subtract $15$ from both sides of the equation.

$- 8 \sqrt{x - 1} = 4 x - 16 - x - 15$

Step 3.2.3

Subtract $x$ from $4 x$ .

$- 8 \sqrt{x - 1} = 3 x - 16 - 15$

Step 3.2.4

Subtract $15$ from $- 16$ .

$- 8 \sqrt{x - 1} = 3 x - 31$

Step 4

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

${(- 8 \sqrt{x - 1})}^{2} = {(3 x - 31)}^{2}$

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

${(- 8 {(x - 1)}^{\frac{1}{2}})}^{2} = {(3 x - 31)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

${(- 8)}^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = {(3 x - 31)}^{2}$

Step 5.2.1.2

Raise $- 8$ to the power of $2$ .

$64 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(3 x - 31)}^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$64 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(3 x - 31)}^{2}$

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.

$64 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(3 x - 31)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$64 {(x - 1)}^{1} = {(3 x - 31)}^{2}$

Step 5.2.1.4

Simplify.

$64 (x - 1) = {(3 x - 31)}^{2}$

Step 5.2.1.5

Apply the distributive property.

$64 x + 64 \cdot - 1 = {(3 x - 31)}^{2}$

Step 5.2.1.6

Multiply $64$ by $- 1$ .

$64 x - 64 = {(3 x - 31)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(3 x - 31)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(3 x - 31)}^{2}$ as $(3 x - 31) (3 x - 31)$ .

$64 x - 64 = (3 x - 31) (3 x - 31)$

Step 5.3.1.2

Expand $(3 x - 31) (3 x - 31)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$64 x - 64 = 3 x (3 x - 31) - 31 (3 x - 31)$

Step 5.3.1.2.2

Apply the distributive property.

$64 x - 64 = 3 x (3 x) + 3 x \cdot - 31 - 31 (3 x - 31)$

Step 5.3.1.2.3

Apply the distributive property.

$64 x - 64 = 3 x (3 x) + 3 x \cdot - 31 - 31 (3 x) - 31 \cdot - 31$

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.

$64 x - 64 = 3 \cdot 3 x \cdot x + 3 x \cdot - 31 - 31 (3 x) - 31 \cdot - 31$

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 $x$ .

$64 x - 64 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 31 - 31 (3 x) - 31 \cdot - 31$

Step 5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$64 x - 64 = 3 \cdot 3 x^{2} + 3 x \cdot - 31 - 31 (3 x) - 31 \cdot - 31$

Step 5.3.1.3.1.3

Multiply $3$ by $3$ .

$64 x - 64 = 9 x^{2} + 3 x \cdot - 31 - 31 (3 x) - 31 \cdot - 31$

Step 5.3.1.3.1.4

Multiply $- 31$ by $3$ .

$64 x - 64 = 9 x^{2} - 93 x - 31 (3 x) - 31 \cdot - 31$

Step 5.3.1.3.1.5

Multiply $3$ by $- 31$ .

$64 x - 64 = 9 x^{2} - 93 x - 93 x - 31 \cdot - 31$

Step 5.3.1.3.1.6

Multiply $- 31$ by $- 31$ .

$64 x - 64 = 9 x^{2} - 93 x - 93 x + 961$

Step 5.3.1.3.2

Subtract $93 x$ from $- 93 x$ .

$64 x - 64 = 9 x^{2} - 186 x + 961$

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.

$9 x^{2} - 186 x + 961 = 64 x - 64$

Step 6.2

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

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Step 6.2.1

Subtract $64 x$ from both sides of the equation.

$9 x^{2} - 186 x + 961 - 64 x = - 64$

Step 6.2.2

Subtract $64 x$ from $- 186 x$ .

$9 x^{2} - 250 x + 961 = - 64$

Step 6.3

Add $64$ to both sides of the equation.

$9 x^{2} - 250 x + 961 + 64 = 0$

Step 6.4

Add $961$ and $64$ .

$9 x^{2} - 250 x + 1025 = 0$

Step 6.5

Factor by grouping.

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Step 6.5.1

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 $a \cdot c = 9 \cdot 1025 = 9225$ and whose sum is $b = - 250$ .

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Step 6.5.1.1

Factor $- 250$ out of $- 250 x$ .

$9 x^{2} - 250 x + 1025 = 0$

Step 6.5.1.2

Rewrite $- 250$ as $- 45$ plus $- 205$

$9 x^{2} + (- 45 - 205) x + 1025 = 0$

Step 6.5.1.3

Apply the distributive property.

$9 x^{2} - 45 x - 205 x + 1025 = 0$

Step 6.5.2

Factor out the greatest common factor from each group.

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Step 6.5.2.1

Group the first two terms and the last two terms.

$(9 x^{2} - 45 x) - 205 x + 1025 = 0$

Step 6.5.2.2

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

$9 x (x - 5) - 205 (x - 5) = 0$

Step 6.5.3

Factor the polynomial by factoring out the greatest common factor, $x - 5$ .

$(x - 5) (9 x - 205) = 0$

Step 6.6

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

$x - 5 = 0$

$9 x - 205 = 0$

Step 6.7

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 6.7.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 6.7.2

Add $5$ to both sides of the equation.

$x = 5$

Step 6.8

Set $9 x - 205$ equal to $0$ and solve for $x$ .

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Step 6.8.1

Set $9 x - 205$ equal to $0$ .

$9 x - 205 = 0$

Step 6.8.2

Solve $9 x - 205 = 0$ for $x$ .

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Step 6.8.2.1

Add $205$ to both sides of the equation.

$9 x = 205$

Step 6.8.2.2

Divide each term in $9 x = 205$ by $9$ and simplify.

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Step 6.8.2.2.1

Divide each term in $9 x = 205$ by $9$ .

$\frac{9 x}{9} = \frac{205}{9}$

Step 6.8.2.2.2

Simplify the left side.

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Step 6.8.2.2.2.1

Cancel the common factor of $9$ .

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Step 6.8.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{205}{9}$

Step 6.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{205}{9}$

Step 6.9

The final solution is all the values that make $(x - 5) (9 x - 205) = 0$ true.

$x = 5, \frac{205}{9}$

Step 7

Exclude the solutions that do not make $2 \sqrt{x - 4} = - \sqrt{x - 1} + 4$ true.

$x = 5$