Solve for x square root of 9x-3 = square root of 5x-5

$\sqrt{9 x} - 3 = \sqrt{5 x - 5}$

Solve for $\sqrt{9 x}$ .

Simplify $\sqrt{5 x - 5}$ .

Rewrite.

$\sqrt{9 x} - 3 = 0 + 0 + \sqrt{5 x - 5}$

Simplify by adding zeros.

$\sqrt{9 x} - 3 = \sqrt{5 x - 5}$

Factor $5$ out of $5 x - 5$ .

Factor $5$ out of $5 x$ .

$\sqrt{9 x} - 3 = \sqrt{5 (x) - 5}$

Factor $5$ out of $- 5$ .

$\sqrt{9 x} - 3 = \sqrt{5 (x) + 5 (- 1)}$

Factor $5$ out of $5 (x) + 5 (- 1)$ .

$\sqrt{9 x} - 3 = \sqrt{5 (x - 1)}$

Add $3$ to both sides of the equation.

$\sqrt{9 x} = \sqrt{5 (x - 1)} + 3$

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

${\sqrt{9 x}}^{2} = {(\sqrt{5 (x - 1)} + 3)}^{2}$

Simplify each side of the equation.

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

${({(9 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{5 (x - 1)} + 3)}^{2}$

Simplify the left side.

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

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

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

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5 (x - 1)} + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5 (x - 1)} + 3)}^{2}$

Rewrite the expression.

${(9 x)}^{1} = {(\sqrt{5 (x - 1)} + 3)}^{2}$

Simplify.

$9 x = {(\sqrt{5 (x - 1)} + 3)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{5 (x - 1)} + 3)}^{2}$ .

Rewrite ${(\sqrt{5 (x - 1)} + 3)}^{2}$ as $(\sqrt{5 (x - 1)} + 3) (\sqrt{5 (x - 1)} + 3)$ .

$9 x = (\sqrt{5 (x - 1)} + 3) (\sqrt{5 (x - 1)} + 3)$

Expand $(\sqrt{5 (x - 1)} + 3) (\sqrt{5 (x - 1)} + 3)$ using the FOIL Method.

Apply the distributive property.

$9 x = \sqrt{5 (x - 1)} (\sqrt{5 (x - 1)} + 3) + 3 (\sqrt{5 (x - 1)} + 3)$

Apply the distributive property.

$9 x = \sqrt{5 (x - 1)} \sqrt{5 (x - 1)} + \sqrt{5 (x - 1)} \cdot 3 + 3 (\sqrt{5 (x - 1)} + 3)$

Apply the distributive property.

$9 x = \sqrt{5 (x - 1)} \sqrt{5 (x - 1)} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{5 (x - 1)} \sqrt{5 (x - 1)}$ .

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

$9 x = {\sqrt{5 (x - 1)}}^{1} \sqrt{5 (x - 1)} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

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

$9 x = {\sqrt{5 (x - 1)}}^{1} {\sqrt{5 (x - 1)}}^{1} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

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

$9 x = {\sqrt{5 (x - 1)}}^{1 + 1} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Add $1$ and $1$ .

$9 x = {\sqrt{5 (x - 1)}}^{2} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Rewrite ${\sqrt{5 (x - 1)}}^{2}$ as $5 (x - 1)$ .

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

$9 x = {({(5 (x - 1))}^{\frac{1}{2}})}^{2} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

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

$9 x = {(5 (x - 1))}^{\frac{1}{2} \cdot 2} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

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

$9 x = {(5 (x - 1))}^{\frac{2}{2}} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x = {(5 (x - 1))}^{\frac{2}{2}} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Rewrite the expression.

$9 x = {(5 (x - 1))}^{1} + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Simplify.

$9 x = 5 (x - 1) + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Apply the distributive property.

$9 x = 5 x + 5 \cdot - 1 + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Multiply $5$ by $- 1$ .

$9 x = 5 x - 5 + \sqrt{5 (x - 1)} \cdot 3 + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Move $3$ to the left of $\sqrt{5 (x - 1)}$ .

$9 x = 5 x - 5 + 3 \cdot \sqrt{5 (x - 1)} + 3 \sqrt{5 (x - 1)} + 3 \cdot 3$

Multiply $3$ by $3$ .

$9 x = 5 x - 5 + 3 \sqrt{5 (x - 1)} + 3 \sqrt{5 (x - 1)} + 9$

Add $- 5$ and $9$ .

$9 x = 5 x + 4 + 3 \sqrt{5 (x - 1)} + 3 \sqrt{5 (x - 1)}$

Add $3 \sqrt{5 (x - 1)}$ and $3 \sqrt{5 (x - 1)}$ .

$9 x = 5 x + 4 + 6 \sqrt{5 (x - 1)}$

Solve for $6 \sqrt{5 (x - 1)}$ .

Rewrite the equation as $5 x + 4 + 6 \sqrt{5 (x - 1)} = 9 x$ .

$5 x + 4 + 6 \sqrt{5 (x - 1)} = 9 x$

Move all terms not containing $6 \sqrt{5 (x - 1)}$ to the right side of the equation.

Subtract $5 x$ from both sides of the equation.

$4 + 6 \sqrt{5 (x - 1)} = 9 x - 5 x$

Subtract $4$ from both sides of the equation.

$6 \sqrt{5 (x - 1)} = 9 x - 5 x - 4$

Subtract $5 x$ from $9 x$ .

$6 \sqrt{5 (x - 1)} = 4 x - 4$

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the distributive property.

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

Simplify the expression.

Multiply $5$ by $- 1$ .

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

Apply the product rule to $6 {(5 x - 5)}^{\frac{1}{2}}$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply.

Multiply $5$ by $36$ .

$180 x + 36 \cdot - 5 = {(4 x - 4)}^{2}$

Multiply $36$ by $- 5$ .

$180 x - 180 = {(4 x - 4)}^{2}$

Simplify the right side.

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

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

$180 x - 180 = (4 x - 4) (4 x - 4)$

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

Apply the distributive property.

$180 x - 180 = 4 x (4 x - 4) - 4 (4 x - 4)$

Apply the distributive property.

$180 x - 180 = 4 x (4 x) + 4 x \cdot - 4 - 4 (4 x - 4)$

Apply the distributive property.

$180 x - 180 = 4 x (4 x) + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$180 x - 180 = 4 \cdot 4 x \cdot x + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

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

Move $x$ .

$180 x - 180 = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Multiply $x$ by $x$ .

$180 x - 180 = 4 \cdot 4 x^{2} + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Multiply $4$ by $4$ .

$180 x - 180 = 16 x^{2} + 4 x \cdot - 4 - 4 (4 x) - 4 \cdot - 4$

Multiply $- 4$ by $4$ .

$180 x - 180 = 16 x^{2} - 16 x - 4 (4 x) - 4 \cdot - 4$

Multiply $4$ by $- 4$ .

$180 x - 180 = 16 x^{2} - 16 x - 16 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$180 x - 180 = 16 x^{2} - 16 x - 16 x + 16$

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

$180 x - 180 = 16 x^{2} - 32 x + 16$

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.

$16 x^{2} - 32 x + 16 = 180 x - 180$

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

Subtract $180 x$ from both sides of the equation.

$16 x^{2} - 32 x + 16 - 180 x = - 180$

Subtract $180 x$ from $- 32 x$ .

$16 x^{2} - 212 x + 16 = - 180$

Add $180$ to both sides of the equation.

$16 x^{2} - 212 x + 16 + 180 = 0$

Add $16$ and $180$ .

$16 x^{2} - 212 x + 196 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} - 212 x + 196$ .

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

$4 (4 x^{2}) - 212 x + 196 = 0$

Factor $4$ out of $- 212 x$ .

$4 (4 x^{2}) + 4 (- 53 x) + 196 = 0$

Factor $4$ out of $196$ .

$4 (4 x^{2}) + 4 (- 53 x) + 4 (49) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (- 53 x)$ .

$4 (4 x^{2} - 53 x) + 4 (49) = 0$

Factor $4$ out of $4 (4 x^{2} - 53 x) + 4 (49)$ .

$4 (4 x^{2} - 53 x + 49) = 0$

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 $a \cdot c = 4 \cdot 49 = 196$ and whose sum is $b = - 53$ .

Factor $- 53$ out of $- 53 x$ .

$4 (4 x^{2} - 53 x + 49) = 0$

Rewrite $- 53$ as $- 4$ plus $- 49$

$4 (4 x^{2} + (- 4 - 49) x + 49) = 0$

Apply the distributive property.

$4 (4 x^{2} - 4 x - 49 x + 49) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} - 4 x) - 49 x + 49) = 0$

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

$4 (4 x (x - 1) - 49 (x - 1)) = 0$

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

$4 ((x - 1) (4 x - 49)) = 0$

Remove unnecessary parentheses.

$4 (x - 1) (4 x - 49) = 0$

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

$x - 1 = 0$

$4 x - 49 = 0$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $4 x - 49$ equal to $0$ and solve for $x$ .

Set $4 x - 49$ equal to $0$ .

$4 x - 49 = 0$

Solve $4 x - 49 = 0$ for $x$ .

Add $49$ to both sides of the equation.

$4 x = 49$

Divide each term in $4 x = 49$ by $4$ and simplify.

Divide each term in $4 x = 49$ by $4$ .

$\frac{4 x}{4} = \frac{49}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{49}{4}$

Divide $x$ by $1$ .

$x = \frac{49}{4}$

The final solution is all the values that make $4 (x - 1) (4 x - 49) = 0$ true.

$x = 1, \frac{49}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = 1, \frac{49}{4}$

Decimal Form:

$x = 1, 12.25$

Mixed Number Form:

$x = 1, 12 \frac{1}{4}$

Solve for x square root of 9x-3 = square root of 5x-5

Related Questions

Support LCMGCF.com by disabling your adblocker.