Solve the Rational Equation for x square root of x^2-5x+4=x-7

$\sqrt{x^{2} - 5 x + 4} = x - 7$

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

${\sqrt{x^{2} - 5 x + 4}}^{2} = {(x - 7)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 5 x + 4}$ as ${(x^{2} - 5 x + 4)}^{\frac{1}{2}}$ .

${({(x^{2} - 5 x + 4)}^{\frac{1}{2}})}^{2} = {(x - 7)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} - 5 x + 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} - 5 x + 4)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} - 5 x + 4)}^{\frac{1}{2} \cdot 2} = {(x - 7)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 5 x + 4)}^{\frac{1}{2} \cdot 2} = {(x - 7)}^{2}$

Rewrite the expression.

${(x^{2} - 5 x + 4)}^{1} = {(x - 7)}^{2}$

Simplify.

$x^{2} - 5 x + 4 = {(x - 7)}^{2}$

Simplify the right side.

Simplify ${(x - 7)}^{2}$ .

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$x^{2} - 5 x + 4 = (x - 7) (x - 7)$

Expand $(x - 7) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 5 x + 4 = x (x - 7) - 7 (x - 7)$

Apply the distributive property.

$x^{2} - 5 x + 4 = x \cdot x + x \cdot - 7 - 7 (x - 7)$

Apply the distributive property.

$x^{2} - 5 x + 4 = x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 5 x + 4 = x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7$

Move $- 7$ to the left of $x$ .

$x^{2} - 5 x + 4 = x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7$

Multiply $- 7$ by $- 7$ .

$x^{2} - 5 x + 4 = x^{2} - 7 x - 7 x + 49$

Subtract $7 x$ from $- 7 x$ .

$x^{2} - 5 x + 4 = x^{2} - 14 x + 49$

Solve for $x$ .

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

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

$x^{2} - 5 x + 4 - x^{2} = - 14 x + 49$

Add $14 x$ to both sides of the equation.

$x^{2} - 5 x + 4 - x^{2} + 14 x = 49$

Combine the opposite terms in $x^{2} - 5 x + 4 - x^{2} + 14 x$ .

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

$- 5 x + 4 + 0 + 14 x = 49$

Add $- 5 x + 4$ and $0$ .

$- 5 x + 4 + 14 x = 49$

Add $- 5 x$ and $14 x$ .

$9 x + 4 = 49$

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

Subtract $4$ from both sides of the equation.

$9 x = 49 - 4$

Subtract $4$ from $49$ .

$9 x = 45$

Divide each term in $9 x = 45$ by $9$ and simplify.

Divide each term in $9 x = 45$ by $9$ .

$\frac{9 x}{9} = \frac{45}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{45}{9}$

Divide $x$ by $1$ .

$x = \frac{45}{9}$

Simplify the right side.

Divide $45$ by $9$ .

$x = 5$

Exclude the solutions that do not make $\sqrt{x^{2} - 5 x + 4} = x - 7$ true.

$No solution$

Solve the Rational Equation for x square root of x^2-5x+4=x-7

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