Solve sqrt{2x+11}-x-4=0

$$\sqrt { 2 x + 11 } - x - 4 = 0$$

Solve for x

$$x=-1$$

Solution Steps

Subtract -x-4 from both sides of the equation.

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

To find the opposite of -x-4, find the opposite of each term.

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

The opposite of -x is x.

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

The opposite of -4 is 4.

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

Square both sides of the equation.

$$\left(\sqrt{2x+11}\right)^{2}=\left(x+4\right)^{2}$$

Calculate \sqrt{2x+11} to the power of 2 and get 2x+11.

$$2x+11=\left(x+4\right)^{2}$$

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

$$2x+11=x^{2}+8x+16$$

Subtract x^{2} from both sides.

$$2x+11-x^{2}=8x+16$$

Subtract 8x from both sides.

$$2x+11-x^{2}-8x=16$$

Combine 2x and -8x to get -6x.

$$-6x+11-x^{2}=16$$

Subtract 16 from both sides.

$$-6x+11-x^{2}-16=0$$

Subtract 16 from 11 to get -5.

$$-6x-5-x^{2}=0$$

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

$$-x^{2}-6x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-\left(-5\right)=5$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-5$$

Rewrite -x^{2}-6x-5 as \left(-x^{2}-x\right)+\left(-5x-5\right).

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

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

$$x\left(-x-1\right)+5\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+5\right)$$

To find equation solutions, solve -x-1=0 and x+5=0.

$$x=-1 x=-5$$

Substitute -1 for x in the equation \sqrt{2x+11}-x-4=0.

$$\sqrt{2\left(-1\right)+11}-\left(-1\right)-4=0$$

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

$$0=0$$

Substitute -5 for x in the equation \sqrt{2x+11}-x-4=0.

$$\sqrt{2\left(-5\right)+11}-\left(-5\right)-4=0$$

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

$$2=0$$

Equation \sqrt{2x+11}=x+4 has a unique solution.

$$x=-1$$