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

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

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

Add $3$ to both sides of the equation.

$\sqrt{2 x - 3} - x + 3 = 0$

Step 2

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

Add $x$ to both sides of the equation.

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

Subtract $3$ from both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$2 x - 3 = (x - 3) (x - 3)$

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

Apply the distributive property.

$2 x - 3 = x (x - 3) - 3 (x - 3)$

Apply the distributive property.

$2 x - 3 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Apply the distributive property.

$2 x - 3 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 3$ by $- 3$ .

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

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

$2 x - 3 = x^{2} - 6 x + 9$

Step 5

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.

$x^{2} - 6 x + 9 = 2 x - 3$

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

Subtract $2 x$ from both sides of the equation.

$x^{2} - 6 x + 9 - 2 x = - 3$

Subtract $2 x$ from $- 6 x$ .

$x^{2} - 8 x + 9 = - 3$

Add $3$ to both sides of the equation.

$x^{2} - 8 x + 9 + 3 = 0$

Add $9$ and $3$ .

$x^{2} - 8 x + 12 = 0$

Factor $x^{2} - 8 x + 12$ 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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Write the factored form using these integers.

$(x - 6) (x - 2) = 0$

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

$x - 6 = 0$

$x - 2 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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 $(x - 6) (x - 2) = 0$ true.

$x = 6, 2$

Step 6

Exclude the solutions that do not make $\sqrt{2 x - 3} - x + 3 = 0$ true.

$x = 6$

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

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