Solve for x square root of 8x-3 = square root of 10x-4

$\sqrt{8 x} - 3 = \sqrt{10 x - 4}$

Solve for $\sqrt{8 x}$ .

Simplify $\sqrt{10 x - 4}$ .

Rewrite.

$\sqrt{8 x} - 3 = 0 + 0 + \sqrt{10 x - 4}$

Simplify by adding zeros.

$\sqrt{8 x} - 3 = \sqrt{10 x - 4}$

Factor $2$ out of $10 x - 4$ .

Factor $2$ out of $10 x$ .

$\sqrt{8 x} - 3 = \sqrt{2 (5 x) - 4}$

Factor $2$ out of $- 4$ .

$\sqrt{8 x} - 3 = \sqrt{2 (5 x) + 2 (- 2)}$

Factor $2$ out of $2 (5 x) + 2 (- 2)$ .

$\sqrt{8 x} - 3 = \sqrt{2 (5 x - 2)}$

Add $3$ to both sides of the equation.

$\sqrt{8 x} = \sqrt{2 (5 x - 2)} + 3$

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

${\sqrt{8 x}}^{2} = {(\sqrt{2 (5 x - 2)} + 3)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x}$ as ${(8 x)}^{\frac{1}{2}}$ .

${({(8 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{2 (5 x - 2)} + 3)}^{2}$

Simplify the left side.

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

Multiply the exponents in ${({(8 x)}^{\frac{1}{2}})}^{2}$ .

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

${(8 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 (5 x - 2)} + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 (5 x - 2)} + 3)}^{2}$

Rewrite the expression.

${(8 x)}^{1} = {(\sqrt{2 (5 x - 2)} + 3)}^{2}$

Simplify.

$8 x = {(\sqrt{2 (5 x - 2)} + 3)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{2 (5 x - 2)} + 3)}^{2}$ .

Rewrite ${(\sqrt{2 (5 x - 2)} + 3)}^{2}$ as $(\sqrt{2 (5 x - 2)} + 3) (\sqrt{2 (5 x - 2)} + 3)$ .

$8 x = (\sqrt{2 (5 x - 2)} + 3) (\sqrt{2 (5 x - 2)} + 3)$

Expand $(\sqrt{2 (5 x - 2)} + 3) (\sqrt{2 (5 x - 2)} + 3)$ using the FOIL Method.

Apply the distributive property.

$8 x = \sqrt{2 (5 x - 2)} (\sqrt{2 (5 x - 2)} + 3) + 3 (\sqrt{2 (5 x - 2)} + 3)$

Apply the distributive property.

$8 x = \sqrt{2 (5 x - 2)} \sqrt{2 (5 x - 2)} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 (\sqrt{2 (5 x - 2)} + 3)$

Apply the distributive property.

$8 x = \sqrt{2 (5 x - 2)} \sqrt{2 (5 x - 2)} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 (5 x - 2)} \sqrt{2 (5 x - 2)}$ .

Raise $\sqrt{2 (5 x - 2)}$ to the power of $1$ .

$8 x = {\sqrt{2 (5 x - 2)}}^{1} \sqrt{2 (5 x - 2)} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Raise $\sqrt{2 (5 x - 2)}$ to the power of $1$ .

$8 x = {\sqrt{2 (5 x - 2)}}^{1} {\sqrt{2 (5 x - 2)}}^{1} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

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

$8 x = {\sqrt{2 (5 x - 2)}}^{1 + 1} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Add $1$ and $1$ .

$8 x = {\sqrt{2 (5 x - 2)}}^{2} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

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

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

$8 x = {({(2 (5 x - 2))}^{\frac{1}{2}})}^{2} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

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

$8 x = {(2 (5 x - 2))}^{\frac{1}{2} \cdot 2} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

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

$8 x = {(2 (5 x - 2))}^{\frac{2}{2}} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$8 x = {(2 (5 x - 2))}^{\frac{2}{2}} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Rewrite the expression.

$8 x = {(2 (5 x - 2))}^{1} + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Simplify.

$8 x = 2 (5 x - 2) + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Apply the distributive property.

$8 x = 2 (5 x) + 2 \cdot - 2 + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Multiply $5$ by $2$ .

$8 x = 10 x + 2 \cdot - 2 + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Multiply $2$ by $- 2$ .

$8 x = 10 x - 4 + \sqrt{2 (5 x - 2)} \cdot 3 + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Move $3$ to the left of $\sqrt{2 (5 x - 2)}$ .

$8 x = 10 x - 4 + 3 \cdot \sqrt{2 (5 x - 2)} + 3 \sqrt{2 (5 x - 2)} + 3 \cdot 3$

Multiply $3$ by $3$ .

$8 x = 10 x - 4 + 3 \sqrt{2 (5 x - 2)} + 3 \sqrt{2 (5 x - 2)} + 9$

Add $- 4$ and $9$ .

$8 x = 10 x + 5 + 3 \sqrt{2 (5 x - 2)} + 3 \sqrt{2 (5 x - 2)}$

Add $3 \sqrt{2 (5 x - 2)}$ and $3 \sqrt{2 (5 x - 2)}$ .

$8 x = 10 x + 5 + 6 \sqrt{2 (5 x - 2)}$

Solve for $6 \sqrt{2 (5 x - 2)}$ .

Rewrite the equation as $10 x + 5 + 6 \sqrt{2 (5 x - 2)} = 8 x$ .

$10 x + 5 + 6 \sqrt{2 (5 x - 2)} = 8 x$

Move all terms not containing $6 \sqrt{2 (5 x - 2)}$ to the right side of the equation.

Subtract $10 x$ from both sides of the equation.

$5 + 6 \sqrt{2 (5 x - 2)} = 8 x - 10 x$

Subtract $5$ from both sides of the equation.

$6 \sqrt{2 (5 x - 2)} = 8 x - 10 x - 5$

Subtract $10 x$ from $8 x$ .

$6 \sqrt{2 (5 x - 2)} = - 2 x - 5$

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

${(6 \sqrt{2 (5 x - 2)})}^{2} = {(- 2 x - 5)}^{2}$

Simplify each side of the equation.

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

${(6 {(2 (5 x - 2))}^{\frac{1}{2}})}^{2} = {(- 2 x - 5)}^{2}$

Simplify the left side.

Simplify ${(6 {(2 (5 x - 2))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(6 {(2 (5 x) + 2 \cdot - 2)}^{\frac{1}{2}})}^{2} = {(- 2 x - 5)}^{2}$

Simplify the expression.

Multiply $5$ by $2$ .

${(6 {(10 x + 2 \cdot - 2)}^{\frac{1}{2}})}^{2} = {(- 2 x - 5)}^{2}$

Multiply $2$ by $- 2$ .

${(6 {(10 x - 4)}^{\frac{1}{2}})}^{2} = {(- 2 x - 5)}^{2}$

Apply the product rule to $6 {(10 x - 4)}^{\frac{1}{2}}$ .

$6^{2} {({(10 x - 4)}^{\frac{1}{2}})}^{2} = {(- 2 x - 5)}^{2}$

Raise $6$ to the power of $2$ .

$36 {({(10 x - 4)}^{\frac{1}{2}})}^{2} = {(- 2 x - 5)}^{2}$

Multiply the exponents in ${({(10 x - 4)}^{\frac{1}{2}})}^{2}$ .

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

$36 {(10 x - 4)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(10 x - 4)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 5)}^{2}$

Rewrite the expression.

$36 {(10 x - 4)}^{1} = {(- 2 x - 5)}^{2}$

Simplify.

$36 (10 x - 4) = {(- 2 x - 5)}^{2}$

Apply the distributive property.

$36 (10 x) + 36 \cdot - 4 = {(- 2 x - 5)}^{2}$

Multiply.

Multiply $10$ by $36$ .

$360 x + 36 \cdot - 4 = {(- 2 x - 5)}^{2}$

Multiply $36$ by $- 4$ .

$360 x - 144 = {(- 2 x - 5)}^{2}$

Simplify the right side.

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

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

$360 x - 144 = (- 2 x - 5) (- 2 x - 5)$

Expand $(- 2 x - 5) (- 2 x - 5)$ using the FOIL Method.

Apply the distributive property.

$360 x - 144 = - 2 x (- 2 x - 5) - 5 (- 2 x - 5)$

Apply the distributive property.

$360 x - 144 = - 2 x (- 2 x) - 2 x \cdot - 5 - 5 (- 2 x - 5)$

Apply the distributive property.

$360 x - 144 = - 2 x (- 2 x) - 2 x \cdot - 5 - 5 (- 2 x) - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$360 x - 144 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 5 - 5 (- 2 x) - 5 \cdot - 5$

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

Move $x$ .

$360 x - 144 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 5 - 5 (- 2 x) - 5 \cdot - 5$

Multiply $x$ by $x$ .

$360 x - 144 = - 2 \cdot - 2 x^{2} - 2 x \cdot - 5 - 5 (- 2 x) - 5 \cdot - 5$

Multiply $- 2$ by $- 2$ .

$360 x - 144 = 4 x^{2} - 2 x \cdot - 5 - 5 (- 2 x) - 5 \cdot - 5$

Multiply $- 5$ by $- 2$ .

$360 x - 144 = 4 x^{2} + 10 x - 5 (- 2 x) - 5 \cdot - 5$

Multiply $- 2$ by $- 5$ .

$360 x - 144 = 4 x^{2} + 10 x + 10 x - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$360 x - 144 = 4 x^{2} + 10 x + 10 x + 25$

Add $10 x$ and $10 x$ .

$360 x - 144 = 4 x^{2} + 20 x + 25$

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.

$4 x^{2} + 20 x + 25 = 360 x - 144$

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

Subtract $360 x$ from both sides of the equation.

$4 x^{2} + 20 x + 25 - 360 x = - 144$

Subtract $360 x$ from $20 x$ .

$4 x^{2} - 340 x + 25 = - 144$

Add $144$ to both sides of the equation.

$4 x^{2} - 340 x + 25 + 144 = 0$

Add $25$ and $144$ .

$4 x^{2} - 340 x + 169 = 0$

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 $a \cdot c = 4 \cdot 169 = 676$ and whose sum is $b = - 340$ .

Factor $- 340$ out of $- 340 x$ .

$4 x^{2} - 340 x + 169 = 0$

Rewrite $- 340$ as $- 2$ plus $- 338$

$4 x^{2} + (- 2 - 338) x + 169 = 0$

Apply the distributive property.

$4 x^{2} - 2 x - 338 x + 169 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} - 2 x) - 338 x + 169 = 0$

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

$2 x (2 x - 1) - 169 (2 x - 1) = 0$

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

$(2 x - 1) (2 x - 169) = 0$

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

$2 x - 1 = 0$

$2 x - 169 = 0$

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 $2 x - 169$ equal to $0$ and solve for $x$ .

Set $2 x - 169$ equal to $0$ .

$2 x - 169 = 0$

Solve $2 x - 169 = 0$ for $x$ .

Add $169$ to both sides of the equation.

$2 x = 169$

Divide each term in $2 x = 169$ by $2$ and simplify.

Divide each term in $2 x = 169$ by $2$ .

$\frac{2 x}{2} = \frac{169}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{169}{2}$

Divide $x$ by $1$ .

$x = \frac{169}{2}$

The final solution is all the values that make $(2 x - 1) (2 x - 169) = 0$ true.

$x = \frac{1}{2}, \frac{169}{2}$

Exclude the solutions that do not make $\sqrt{8 x} - 3 = \sqrt{10 x - 4}$ true.

$No solution$

Solve for x square root of 8x-3 = square root of 10x-4

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