Solve x-1=sqrt{x+11}

$$x-1 = \sqrt{ x+11 }$$

Solve for x

$$x=5$$

Solution Steps

Square both sides of the equation.

$$\left(x-1\right)^{2}=\left(\sqrt{x+11}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=\left(\sqrt{x+11}\right)^{2}$$

Calculate \sqrt{x+11} to the power of 2 and get x+11.

$$x^{2}-2x+1=x+11$$

Subtract x from both sides.

$$x^{2}-2x+1-x=11$$

Combine -2x and -x to get -3x.

$$x^{2}-3x+1=11$$

Subtract 11 from both sides.

$$x^{2}-3x+1-11=0$$

Subtract 11 from 1 to get -10.

$$x^{2}-3x-10=0$$

To solve the equation, factor x^{2}-3x-10 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-3 ab=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -3.

$$a=-5 b=2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-5\right)\left(x+2\right)$$

To find equation solutions, solve x-5=0 and x+2=0.

$$x=5 x=-2$$

Substitute 5 for x in the equation x-1=\sqrt{x+11}.

$$5-1=\sqrt{5+11}$$

Simplify. The value x=5 satisfies the equation.

$$4=4$$

Substitute -2 for x in the equation x-1=\sqrt{x+11}.

$$-2-1=\sqrt{-2+11}$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-3=3$$

Equation x-1=\sqrt{x+11} has a unique solution.

$$x=5$$