Solve x-4=sqrt{x}+2

$$x-4= \sqrt{ x } +2$$

Solve for x

$$x=9$$

Solution Steps

Subtract 2 from both sides of the equation.

$$x-4-2=\sqrt{x}$$

Subtract 2 from -4 to get -6.

$$x-6=\sqrt{x}$$

Square both sides of the equation.

$$\left(x-6\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36=\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}-12x+36=x$$

Subtract x from both sides.

$$x^{2}-12x+36-x=0$$

Combine -12x and -x to get -13x.

$$x^{2}-13x+36=0$$

To solve the equation, factor x^{2}-13x+36 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.

$$a+b=-13 ab=36$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -13.

$$a=-9 b=-4$$

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

$$\left(x-9\right)\left(x-4\right)$$

To find equation solutions, solve x-9=0 and x-4=0.

$$x=9 x=4$$

Substitute 9 for x in the equation x-4=\sqrt{x}+2.

$$9-4=\sqrt{9}+2$$

Simplify. The value x=9 satisfies the equation.

$$5=5$$

Substitute 4 for x in the equation x-4=\sqrt{x}+2.

$$4-4=\sqrt{4}+2$$

Simplify. The value x=4 does not satisfy the equation.

$$0=4$$

Equation x-6=\sqrt{x} has a unique solution.

$$x=9$$