Solve x-8=sqrt{x-6}

$$x - 8 = \sqrt { x - 6 }$$

Solve for x

$$x=10$$

Solution Steps

Square both sides of the equation.

$$\left(x-8\right)^{2}=\left(\sqrt{x-6}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64=\left(\sqrt{x-6}\right)^{2}$$

Calculate \sqrt{x-6} to the power of 2 and get x-6.

$$x^{2}-16x+64=x-6$$

Subtract x from both sides.

$$x^{2}-16x+64-x=-6$$

Combine -16x and -x to get -17x.

$$x^{2}-17x+64=-6$$

Add 6 to both sides.

$$x^{2}-17x+64+6=0$$

Add 64 and 6 to get 70.

$$x^{2}-17x+70=0$$

To solve the equation, factor x^{2}-17x+70 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.

$$a+b=-17 ab=70$$

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 70.

$$-1,-70 -2,-35 -5,-14 -7,-10$$

Calculate the sum for each pair.

$$-1-70=-71 -2-35=-37 -5-14=-19 -7-10=-17$$

The solution is the pair that gives sum -17.

$$a=-10 b=-7$$

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

$$\left(x-10\right)\left(x-7\right)$$

To find equation solutions, solve x-10=0 and x-7=0.

$$x=10 x=7$$

Substitute 10 for x in the equation x-8=\sqrt{x-6}.

$$10-8=\sqrt{10-6}$$

Simplify. The value x=10 satisfies the equation.

$$2=2$$

Substitute 7 for x in the equation x-8=\sqrt{x-6}.

$$7-8=\sqrt{7-6}$$

Simplify. The value x=7 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation x-8=\sqrt{x-6} has a unique solution.

$$x=10$$