Solve x-sqrt{2x-6}=7

$$x - \sqrt { 2 x - 6 } = 7$$

Solve for x

$$x=11$$

Solution Steps

Subtract x from both sides of the equation.

$$-\sqrt{2x-6}=7-x$$

Square both sides of the equation.

$$\left(-\sqrt{2x-6}\right)^{2}=\left(7-x\right)^{2}$$

Expand \left(-\sqrt{2x-6}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{2x-6}\right)^{2}=\left(7-x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{2x-6}\right)^{2}=\left(7-x\right)^{2}$$

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

$$1\left(2x-6\right)=\left(7-x\right)^{2}$$

Use the distributive property to multiply 1 by 2x-6.

$$2x-6=\left(7-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-x\right)^{2}.

$$2x-6=49-14x+x^{2}$$

Subtract 49 from both sides.

$$2x-6-49=-14x+x^{2}$$

Subtract 49 from -6 to get -55.

$$2x-55=-14x+x^{2}$$

Add 14x to both sides.

$$2x-55+14x=x^{2}$$

Combine 2x and 14x to get 16x.

$$16x-55=x^{2}$$

Subtract x^{2} from both sides.

$$16x-55-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+16x-55=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-55. To find a and b, set up a system to be solved.

$$a+b=16 ab=-\left(-55\right)=55$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 55.

$$1,55 5,11$$

Calculate the sum for each pair.

$$1+55=56 5+11=16$$

The solution is the pair that gives sum 16.

$$a=11 b=5$$

Rewrite -x^{2}+16x-55 as \left(-x^{2}+11x\right)+\left(5x-55\right).

$$\left(-x^{2}+11x\right)+\left(5x-55\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-11\right)+5\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(-x+5\right)$$

To find equation solutions, solve x-11=0 and -x+5=0.

$$x=11 x=5$$

Substitute 11 for x in the equation x-\sqrt{2x-6}=7.

$$11-\sqrt{2\times 11-6}=7$$

Simplify. The value x=11 satisfies the equation.

$$7=7$$

Substitute 5 for x in the equation x-\sqrt{2x-6}=7.

$$5-\sqrt{2\times 5-6}=7$$

Simplify. The value x=5 does not satisfy the equation.

$$3=7$$

Equation -\sqrt{2x-6}=7-x has a unique solution.

$$x=11$$