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

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

Solve for x

$$x=-3$$

Solution Steps

Subtract -4 from both sides of the equation.

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

Add 1 and 4 to get 5.

$$\sqrt{1-x}=x+5$$

Square both sides of the equation.

$$\left(\sqrt{1-x}\right)^{2}=\left(x+5\right)^{2}$$

Calculate \sqrt{1-x} to the power of 2 and get 1-x.

$$1-x=\left(x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$1-x=x^{2}+10x+25$$

Subtract x^{2} from both sides.

$$1-x-x^{2}=10x+25$$

Subtract 10x from both sides.

$$1-x-x^{2}-10x=25$$

Combine -x and -10x to get -11x.

$$1-11x-x^{2}=25$$

Subtract 25 from both sides.

$$1-11x-x^{2}-25=0$$

Subtract 25 from 1 to get -24.

$$-24-11x-x^{2}=0$$

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

$$-x^{2}-11x-24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-11 ab=-\left(-24\right)=24$$

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 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-3 b=-8$$

Rewrite -x^{2}-11x-24 as \left(-x^{2}-3x\right)+\left(-8x-24\right).

$$\left(-x^{2}-3x\right)+\left(-8x-24\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x-3\right)+8\left(-x-3\right)$$

Factor out common term -x-3 by using distributive property.

$$\left(-x-3\right)\left(x+8\right)$$

To find equation solutions, solve -x-3=0 and x+8=0.

$$x=-3 x=-8$$

Substitute -3 for x in the equation \sqrt{1-x}-4=x+1.

$$\sqrt{1-\left(-3\right)}-4=-3+1$$

Simplify. The value x=-3 satisfies the equation.

$$-2=-2$$

Substitute -8 for x in the equation \sqrt{1-x}-4=x+1.

$$\sqrt{1-\left(-8\right)}-4=-8+1$$

Simplify. The value x=-8 does not satisfy the equation.

$$-1=-7$$

Equation \sqrt{1-x}=x+5 has a unique solution.

$$x=-3$$