Solve x=sqrt{6x-26}+3

$$x = \sqrt { 6 x - 26 } + 3$$

Solve for x

$$x=7$$

$$x=5$$

Solution Steps

Subtract 3 from both sides of the equation.

$$x-3=\sqrt{6x-26}$$

Square both sides of the equation.

$$\left(x-3\right)^{2}=\left(\sqrt{6x-26}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=\left(\sqrt{6x-26}\right)^{2}$$

Calculate \sqrt{6x-26} to the power of 2 and get 6x-26.

$$x^{2}-6x+9=6x-26$$

Subtract 6x from both sides.

$$x^{2}-6x+9-6x=-26$$

Combine -6x and -6x to get -12x.

$$x^{2}-12x+9=-26$$

Add 26 to both sides.

$$x^{2}-12x+9+26=0$$

Add 9 and 26 to get 35.

$$x^{2}-12x+35=0$$

To solve the equation, factor x^{2}-12x+35 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.

$$a+b=-12 ab=35$$

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 35.

$$-1,-35 -5,-7$$

Calculate the sum for each pair.

$$-1-35=-36 -5-7=-12$$

The solution is the pair that gives sum -12.

$$a=-7 b=-5$$

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

$$\left(x-7\right)\left(x-5\right)$$

To find equation solutions, solve x-7=0 and x-5=0.

$$x=7 x=5$$

Substitute 7 for x in the equation x=\sqrt{6x-26}+3.

$$7=\sqrt{6\times 7-26}+3$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

Substitute 5 for x in the equation x=\sqrt{6x-26}+3.

$$5=\sqrt{6\times 5-26}+3$$

Simplify. The value x=5 satisfies the equation.

$$5=5$$

List all solutions of x-3=\sqrt{6x-26}.

$$x=7 x=5$$