Solve by Factoring square root of 2x-5- square root of x-3=1

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

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{2 x - 5} - \sqrt{x - 3} - 1 = 0$

Step 2

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

Add $\sqrt{x - 3}$ to both sides of the equation.

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

Add $1$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x - 3} \sqrt{x - 3}$ .

Raise $\sqrt{x - 3}$ to the power of $1$ .

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

Raise $\sqrt{x - 3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x - 3}}^{2}$ as $x - 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $\sqrt{x - 3}$ by $1$ .

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

Multiply $\sqrt{x - 3}$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $- 3$ and $1$ .

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

Add $\sqrt{x - 3}$ and $\sqrt{x - 3}$ .

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

Step 5

Solve for $2 \sqrt{x - 3}$ .

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

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

Move all terms not containing $2 \sqrt{x - 3}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

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

Add $2$ to both sides of the equation.

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

Subtract $x$ from $2 x$ .

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

Add $- 5$ and $2$ .

$2 \sqrt{x - 3} = x - 3$

Step 6

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

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

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(2 {(x - 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(x - 3)}^{\frac{1}{2}}$ .

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

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

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

Multiply the exponents in ${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4 {(x - 3)}^{1} = {(x - 3)}^{2}$

Simplify.

$4 (x - 3) = {(x - 3)}^{2}$

Apply the distributive property.

$4 x + 4 \cdot - 3 = {(x - 3)}^{2}$

Multiply $4$ by $- 3$ .

$4 x - 12 = {(x - 3)}^{2}$

Simplify the right side.

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

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

$4 x - 12 = (x - 3) (x - 3)$

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

Apply the distributive property.

$4 x - 12 = x (x - 3) - 3 (x - 3)$

Apply the distributive property.

$4 x - 12 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Apply the distributive property.

$4 x - 12 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x - 12 = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3$

Move $- 3$ to the left of $x$ .

$4 x - 12 = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$4 x - 12 = x^{2} - 3 x - 3 x + 9$

Subtract $3 x$ from $- 3 x$ .

$4 x - 12 = x^{2} - 6 x + 9$

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.

$x^{2} - 6 x + 9 = 4 x - 12$

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

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

$x^{2} - 6 x + 9 - 4 x = - 12$

Subtract $4 x$ from $- 6 x$ .

$x^{2} - 10 x + 9 = - 12$

Add $12$ to both sides of the equation.

$x^{2} - 10 x + 9 + 12 = 0$

Add $9$ and $12$ .

$x^{2} - 10 x + 21 = 0$

Factor $x^{2} - 10 x + 21$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

$(x - 7) (x - 3) = 0$

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

$x - 7 = 0$

$x - 3 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(x - 7) (x - 3) = 0$ true.

$x = 7, 3$

Solve by Factoring square root of 2x-5- square root of x-3=1

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