Solve for x square root of 2x-1 = square root of 8x-4

$\sqrt{2 x} - 1 = \sqrt{8 x - 4}$

Solve for $\sqrt{2 x}$ .

Simplify $\sqrt{8 x - 4}$ .

Rewrite.

$\sqrt{2 x} - 1 = 0 + 0 + \sqrt{8 x - 4}$

Simplify by adding zeros.

$\sqrt{2 x} - 1 = \sqrt{8 x - 4}$

Factor $4$ out of $8 x - 4$ .

Factor $4$ out of $8 x$ .

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

Factor $4$ out of $- 4$ .

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

Factor $4$ out of $4 (2 x) + 4 (- 1)$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Add $1$ to both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2 \sqrt{2 x - 1} (2 \sqrt{2 x - 1})$ .

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $2$ by $4$ .

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

Multiply $4$ by $- 1$ .

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

Multiply $2$ by $1$ .

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

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

$2 x = 8 x - 4 + 2 \sqrt{2 x - 1} + 2 \sqrt{2 x - 1} + 1 \cdot 1$

Multiply $1$ by $1$ .

$2 x = 8 x - 4 + 2 \sqrt{2 x - 1} + 2 \sqrt{2 x - 1} + 1$

Add $- 4$ and $1$ .

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

Add $2 \sqrt{2 x - 1}$ and $2 \sqrt{2 x - 1}$ .

$2 x = 8 x - 3 + 4 \sqrt{2 x - 1}$

Solve for $4 \sqrt{2 x - 1}$ .

Rewrite the equation as $8 x - 3 + 4 \sqrt{2 x - 1} = 2 x$ .

$8 x - 3 + 4 \sqrt{2 x - 1} = 2 x$

Move all terms not containing $4 \sqrt{2 x - 1}$ to the right side of the equation.

Subtract $8 x$ from both sides of the equation.

$- 3 + 4 \sqrt{2 x - 1} = 2 x - 8 x$

Add $3$ to both sides of the equation.

$4 \sqrt{2 x - 1} = 2 x - 8 x + 3$

Subtract $8 x$ from $2 x$ .

$4 \sqrt{2 x - 1} = - 6 x + 3$

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

${(4 \sqrt{2 x - 1})}^{2} = {(- 6 x + 3)}^{2}$

Simplify each side of the equation.

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

${(4 {(2 x - 1)}^{\frac{1}{2}})}^{2} = {(- 6 x + 3)}^{2}$

Simplify the left side.

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

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

$4^{2} {({(2 x - 1)}^{\frac{1}{2}})}^{2} = {(- 6 x + 3)}^{2}$

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

$16 {({(2 x - 1)}^{\frac{1}{2}})}^{2} = {(- 6 x + 3)}^{2}$

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

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

$16 {(2 x - 1)}^{\frac{1}{2} \cdot 2} = {(- 6 x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(2 x - 1)}^{\frac{1}{2} \cdot 2} = {(- 6 x + 3)}^{2}$

Rewrite the expression.

$16 {(2 x - 1)}^{1} = {(- 6 x + 3)}^{2}$

Simplify.

$16 (2 x - 1) = {(- 6 x + 3)}^{2}$

Apply the distributive property.

$16 (2 x) + 16 \cdot - 1 = {(- 6 x + 3)}^{2}$

Multiply.

Multiply $2$ by $16$ .

$32 x + 16 \cdot - 1 = {(- 6 x + 3)}^{2}$

Multiply $16$ by $- 1$ .

$32 x - 16 = {(- 6 x + 3)}^{2}$

Simplify the right side.

Simplify ${(- 6 x + 3)}^{2}$ .

Rewrite ${(- 6 x + 3)}^{2}$ as $(- 6 x + 3) (- 6 x + 3)$ .

$32 x - 16 = (- 6 x + 3) (- 6 x + 3)$

Expand $(- 6 x + 3) (- 6 x + 3)$ using the FOIL Method.

Apply the distributive property.

$32 x - 16 = - 6 x (- 6 x + 3) + 3 (- 6 x + 3)$

Apply the distributive property.

$32 x - 16 = - 6 x (- 6 x) - 6 x \cdot 3 + 3 (- 6 x + 3)$

Apply the distributive property.

$32 x - 16 = - 6 x (- 6 x) - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$32 x - 16 = - 6 \cdot - 6 x \cdot x - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

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

Move $x$ .

$32 x - 16 = - 6 \cdot - 6 (x \cdot x) - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Multiply $x$ by $x$ .

$32 x - 16 = - 6 \cdot - 6 x^{2} - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Multiply $- 6$ by $- 6$ .

$32 x - 16 = 36 x^{2} - 6 x \cdot 3 + 3 (- 6 x) + 3 \cdot 3$

Multiply $3$ by $- 6$ .

$32 x - 16 = 36 x^{2} - 18 x + 3 (- 6 x) + 3 \cdot 3$

Multiply $- 6$ by $3$ .

$32 x - 16 = 36 x^{2} - 18 x - 18 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$32 x - 16 = 36 x^{2} - 18 x - 18 x + 9$

Subtract $18 x$ from $- 18 x$ .

$32 x - 16 = 36 x^{2} - 36 x + 9$

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.

$36 x^{2} - 36 x + 9 = 32 x - 16$

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

Subtract $32 x$ from both sides of the equation.

$36 x^{2} - 36 x + 9 - 32 x = - 16$

Subtract $32 x$ from $- 36 x$ .

$36 x^{2} - 68 x + 9 = - 16$

Add $16$ to both sides of the equation.

$36 x^{2} - 68 x + 9 + 16 = 0$

Add $9$ and $16$ .

$36 x^{2} - 68 x + 25 = 0$

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 $a \cdot c = 36 \cdot 25 = 900$ and whose sum is $b = - 68$ .

Factor $- 68$ out of $- 68 x$ .

$36 x^{2} - 68 x + 25 = 0$

Rewrite $- 68$ as $- 18$ plus $- 50$

$36 x^{2} + (- 18 - 50) x + 25 = 0$

Apply the distributive property.

$36 x^{2} - 18 x - 50 x + 25 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(36 x^{2} - 18 x) - 50 x + 25 = 0$

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

$18 x (2 x - 1) - 25 (2 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x - 1) (18 x - 25) = 0$

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

$2 x - 1 = 0$

$18 x - 25 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Set $18 x - 25$ equal to $0$ and solve for $x$ .

Set $18 x - 25$ equal to $0$ .

$18 x - 25 = 0$

Solve $18 x - 25 = 0$ for $x$ .

Add $25$ to both sides of the equation.

$18 x = 25$

Divide each term in $18 x = 25$ by $18$ and simplify.

Divide each term in $18 x = 25$ by $18$ .

$\frac{18 x}{18} = \frac{25}{18}$

Simplify the left side.

Cancel the common factor of $18$ .

Cancel the common factor.

$\frac{18 x}{18} = \frac{25}{18}$

Divide $x$ by $1$ .

$x = \frac{25}{18}$

The final solution is all the values that make $(2 x - 1) (18 x - 25) = 0$ true.

$x = \frac{1}{2}, \frac{25}{18}$

Exclude the solutions that do not make $\sqrt{2 x} - 1 = \sqrt{8 x - 4}$ true.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$

Solve for x square root of 2x-1 = square root of 8x-4

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