Solve for x square root of x^2-3x-6=x-1

$\sqrt{x^{2} - 3 x - 6} = x - 1$

Step 1

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

${\sqrt{x^{2} - 3 x - 6}}^{2} = {(x - 1)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 3 x - 6}$ as ${(x^{2} - 3 x - 6)}^{\frac{1}{2}}$ .

${({(x^{2} - 3 x - 6)}^{\frac{1}{2}})}^{2} = {(x - 1)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} - 3 x - 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} - 3 x - 6)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} - 3 x - 6)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 3 x - 6)}^{\frac{1}{2} \cdot 2} = {(x - 1)}^{2}$

Rewrite the expression.

${(x^{2} - 3 x - 6)}^{1} = {(x - 1)}^{2}$

Simplify.

$x^{2} - 3 x - 6 = {(x - 1)}^{2}$

Simplify the right side.

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

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

$x^{2} - 3 x - 6 = (x - 1) (x - 1)$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 3 x - 6 = x (x - 1) - 1 (x - 1)$

Apply the distributive property.

$x^{2} - 3 x - 6 = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Apply the distributive property.

$x^{2} - 3 x - 6 = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 3 x - 6 = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Move $- 1$ to the left of $x$ .

$x^{2} - 3 x - 6 = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 3 x - 6 = x^{2} - x - 1 x - 1 \cdot - 1$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 3 x - 6 = x^{2} - x - x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$x^{2} - 3 x - 6 = x^{2} - x - x + 1$

Subtract $x$ from $- x$ .

$x^{2} - 3 x - 6 = x^{2} - 2 x + 1$

Step 3

Solve for $x$ .

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

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

$x^{2} - 3 x - 6 - x^{2} = - 2 x + 1$

Add $2 x$ to both sides of the equation.

$x^{2} - 3 x - 6 - x^{2} + 2 x = 1$

Combine the opposite terms in $x^{2} - 3 x - 6 - x^{2} + 2 x$ .

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

$- 3 x - 6 + 0 + 2 x = 1$

Add $- 3 x - 6$ and $0$ .

$- 3 x - 6 + 2 x = 1$

Add $- 3 x$ and $2 x$ .

$- x - 6 = 1$

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

Add $6$ to both sides of the equation.

$- x = 1 + 6$

Add $1$ and $6$ .

$- x = 7$

Divide each term in $- x = 7$ by $- 1$ and simplify.

Divide each term in $- x = 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{7}{- 1}$

Divide $x$ by $1$ .

$x = \frac{7}{- 1}$

Simplify the right side.

Divide $7$ by $- 1$ .

$x = - 7$

Step 4

Exclude the solutions that do not make $\sqrt{x^{2} - 3 x - 6} = x - 1$ true.

$No solution$

Solve for x square root of x^2-3x-6=x-1

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