Solve by Factoring square root of 5x-1- square root of x+2=1

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

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{5 x - 1} - \sqrt{x + 2} - 1 = 0$

Step 2

Move all terms not containing $\sqrt{5 x - 1}$ to the right side of the equation.

Add $\sqrt{x + 2}$ to both sides of the equation.

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

Add $1$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 2} \sqrt{x + 2}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x + 2}}^{2}$ as $x + 2$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $\sqrt{x + 2}$ by $1$ .

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

Multiply $\sqrt{x + 2}$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $2$ and $1$ .

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

Add $\sqrt{x + 2}$ and $\sqrt{x + 2}$ .

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

Step 5

Solve for $2 \sqrt{x + 2}$ .

Rewrite the equation as $x + 3 + 2 \sqrt{x + 2} = 5 x - 1$ .

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $3$ from both sides of the equation.

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

Subtract $x$ from $5 x$ .

$2 \sqrt{x + 2} = 4 x - 1 - 3$

Subtract $3$ from $- 1$ .

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

Step 6

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

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

Step 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $2$ .

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

Simplify the right side.

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

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

$4 x + 8 = (4 x - 4) (4 x - 4)$

Expand $(4 x - 4) (4 x - 4)$ using the FOIL Method.

Apply the distributive property.

$4 x + 8 = 4 x (4 x - 4) - 4 (4 x - 4)$

Apply the distributive property.

$4 x + 8 = 4 x (4 x) + 4 x \cdot - 4 - 4 (4 x - 4)$

Apply the distributive property.

$4 x + 8 = 4 x (4 x) + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 x + 8 = 4 \cdot 4 x \cdot x + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

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

Move $x$ .

$4 x + 8 = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Multiply $x$ by $x$ .

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

Multiply $4$ by $4$ .

$4 x + 8 = 16 x^{2} + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Multiply $- 4$ by $4$ .

$4 x + 8 = 16 x^{2} - 16 x - 4 (4 x) - 4 \cdot - 4$

Multiply $4$ by $- 4$ .

$4 x + 8 = 16 x^{2} - 16 x - 16 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$4 x + 8 = 16 x^{2} - 16 x - 16 x + 16$

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

$4 x + 8 = 16 x^{2} - 32 x + 16$

Step 8

Solve for $x$ .

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

$16 x^{2} - 32 x + 16 = 4 x + 8$

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

Subtract $4 x$ from both sides of the equation.

$16 x^{2} - 32 x + 16 - 4 x = 8$

Subtract $4 x$ from $- 32 x$ .

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

Subtract $8$ from both sides of the equation.

$16 x^{2} - 36 x + 16 - 8 = 0$

Subtract $8$ from $16$ .

$16 x^{2} - 36 x + 8 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} - 36 x + 8$ .

Factor $4$ out of $16 x^{2}$ .

$4 (4 x^{2}) - 36 x + 8 = 0$

Factor $4$ out of $- 36 x$ .

$4 (4 x^{2}) + 4 (- 9 x) + 8 = 0$

Factor $4$ out of $8$ .

$4 (4 x^{2}) + 4 (- 9 x) + 4 (2) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (- 9 x)$ .

$4 (4 x^{2} - 9 x) + 4 (2) = 0$

Factor $4$ out of $4 (4 x^{2} - 9 x) + 4 (2)$ .

$4 (4 x^{2} - 9 x + 2) = 0$

Factor.

Factor by grouping.

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 $a \cdot c = 4 \cdot 2 = 8$ and whose sum is $b = - 9$ .

Factor $- 9$ out of $- 9 x$ .

$4 (4 x^{2} - 9 x + 2) = 0$

Rewrite $- 9$ as $- 1$ plus $- 8$

$4 (4 x^{2} + (- 1 - 8) x + 2) = 0$

Apply the distributive property.

$4 (4 x^{2} - 1 x - 8 x + 2) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} - 1 x) - 8 x + 2) = 0$

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

$4 (x (4 x - 1) - 2 (4 x - 1)) = 0$

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

$4 ((4 x - 1) (x - 2)) = 0$

Remove unnecessary parentheses.

$4 (4 x - 1) (x - 2) = 0$

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

$4 x - 1 = 0$

$x - 2 = 0$

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

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

$4 x - 1 = 0$

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

Add $1$ to both sides of the equation.

$4 x = 1$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $4 (4 x - 1) (x - 2) = 0$ true.

$x = \frac{1}{4}, 2$

Step 9

Exclude the solutions that do not make $\sqrt{5 x - 1} - \sqrt{x + 2} - 1 = 0$ true.

$x = 2$

Solve by Factoring square root of 5x-1- square root of x+2=1

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