Solve the Rational Equation for x square root of x^2-5=x-2

$\sqrt{x^{2} - 5} = x - 2$

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

$x^{2} - 5 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$x^{2} - 5 = x^{2} - 2 x - 2 x + 4$

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

$x^{2} - 5 = x^{2} - 4 x + 4$

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.

$x^{2} - 4 x + 4 = x^{2} - 5$

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

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 4 x + 4 - x^{2} = - 5$

Combine the opposite terms in $x^{2} - 4 x + 4 - x^{2}$ .

Subtract $x^{2}$ from $x^{2}$ .

$- 4 x + 4 + 0 = - 5$

Add $- 4 x + 4$ and $0$ .

$- 4 x + 4 = - 5$

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$- 4 x = - 5 - 4$

Subtract $4$ from $- 5$ .

$- 4 x = - 9$

Divide each term in $- 4 x = - 9$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 9$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 9}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 9}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 9}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{9}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{4}$

Decimal Form:

$x = 2.25$

Mixed Number Form:

$x = 2 \frac{1}{4}$

Solve the Rational Equation for x square root of x^2-5=x-2

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