Solve 4x+5-sqrt{x+1}=1

$$4 x + 5 - \sqrt { x + 1 } = 1$$

Solve for x

$$x=-\frac{15}{16}=-0.9375$$

$$x=-1$$

Solution Steps

Subtract 4x+5 from both sides of the equation.

$$-\sqrt{x+1}=1-\left(4x+5\right)$$

To find the opposite of 4x+5, find the opposite of each term.

$$-\sqrt{x+1}=1-4x-5$$

Subtract 5 from 1 to get -4.

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

Square both sides of the equation.

$$\left(-\sqrt{x+1}\right)^{2}=\left(-4-4x\right)^{2}$$

Expand \left(-\sqrt{x+1}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{x+1}\right)^{2}=\left(-4-4x\right)^{2}$$

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

$$1\left(\sqrt{x+1}\right)^{2}=\left(-4-4x\right)^{2}$$

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

$$1\left(x+1\right)=\left(-4-4x\right)^{2}$$

Use the distributive property to multiply 1 by x+1.

$$x+1=\left(-4-4x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-4-4x\right)^{2}.

$$x+1=16+32x+16x^{2}$$

Subtract 16 from both sides.

$$x+1-16=32x+16x^{2}$$

Subtract 16 from 1 to get -15.

$$x-15=32x+16x^{2}$$

Subtract 32x from both sides.

$$x-15-32x=16x^{2}$$

Combine x and -32x to get -31x.

$$-31x-15=16x^{2}$$

Subtract 16x^{2} from both sides.

$$-31x-15-16x^{2}=0$$

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

$$-16x^{2}-31x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -16x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-31 ab=-16\left(-15\right)=240$$

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 240.

$$-1,-240 -2,-120 -3,-80 -4,-60 -5,-48 -6,-40 -8,-30 -10,-24 -12,-20 -15,-16$$

Calculate the sum for each pair.

$$-1-240=-241 -2-120=-122 -3-80=-83 -4-60=-64 -5-48=-53 -6-40=-46 -8-30=-38 -10-24=-34 -12-20=-32 -15-16=-31$$

The solution is the pair that gives sum -31.

$$a=-15 b=-16$$

Rewrite -16x^{2}-31x-15 as \left(-16x^{2}-15x\right)+\left(-16x-15\right).

$$\left(-16x^{2}-15x\right)+\left(-16x-15\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(16x+15\right)-\left(16x+15\right)$$

Factor out common term 16x+15 by using distributive property.

$$\left(16x+15\right)\left(-x-1\right)$$

To find equation solutions, solve 16x+15=0 and -x-1=0.

$$x=-\frac{15}{16} x=-1$$

Substitute -\frac{15}{16} for x in the equation 4x+5-\sqrt{x+1}=1.

$$4\left(-\frac{15}{16}\right)+5-\sqrt{-\frac{15}{16}+1}=1$$

Simplify. The value x=-\frac{15}{16} satisfies the equation.

$$1=1$$

Substitute -1 for x in the equation 4x+5-\sqrt{x+1}=1.

$$4\left(-1\right)+5-\sqrt{-1+1}=1$$

Simplify. The value x=-1 satisfies the equation.

$$1=1$$

List all solutions of -\sqrt{x+1}=-4x-4.

$$x=-\frac{15}{16} x=-1$$