Solve 1+sqrt{4-2x}=x+11

$$1 + \sqrt { 4 - 2 x } = x + 11$$

Solve for x

$$x=-6$$

Solution Steps

Subtract 1 from both sides of the equation.

$$\sqrt{4-2x}=x+11-1$$

Subtract 1 from 11 to get 10.

$$\sqrt{4-2x}=x+10$$

Square both sides of the equation.

$$\left(\sqrt{4-2x}\right)^{2}=\left(x+10\right)^{2}$$

Calculate \sqrt{4-2x} to the power of 2 and get 4-2x.

$$4-2x=\left(x+10\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+10\right)^{2}.

$$4-2x=x^{2}+20x+100$$

Subtract x^{2} from both sides.

$$4-2x-x^{2}=20x+100$$

Subtract 20x from both sides.

$$4-2x-x^{2}-20x=100$$

Combine -2x and -20x to get -22x.

$$4-22x-x^{2}=100$$

Subtract 100 from both sides.

$$4-22x-x^{2}-100=0$$

Subtract 100 from 4 to get -96.

$$-96-22x-x^{2}=0$$

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

$$-x^{2}-22x-96=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-96. To find a and b, set up a system to be solved.

$$a+b=-22 ab=-\left(-96\right)=96$$

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 96.

$$-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12$$

Calculate the sum for each pair.

$$-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20$$

The solution is the pair that gives sum -22.

$$a=-6 b=-16$$

Rewrite -x^{2}-22x-96 as \left(-x^{2}-6x\right)+\left(-16x-96\right).

$$\left(-x^{2}-6x\right)+\left(-16x-96\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(-x-6\right)+16\left(-x-6\right)$$

Factor out common term -x-6 by using distributive property.

$$\left(-x-6\right)\left(x+16\right)$$

To find equation solutions, solve -x-6=0 and x+16=0.

$$x=-6 x=-16$$

Substitute -6 for x in the equation 1+\sqrt{4-2x}=x+11.

$$1+\sqrt{4-2\left(-6\right)}=-6+11$$

Simplify. The value x=-6 satisfies the equation.

$$5=5$$

Substitute -16 for x in the equation 1+\sqrt{4-2x}=x+11.

$$1+\sqrt{4-2\left(-16\right)}=-16+11$$

Simplify. The value x=-16 does not satisfy the equation because the left and the right hand side have opposite signs.

$$7=-5$$

Equation \sqrt{4-2x}=x+10 has a unique solution.

$$x=-6$$