Solve sqrt{x+15}+1=x+4

$$\sqrt { x + 15 } + 1 = x + 4$$

Solve for x

$$x=1$$

Solution Steps

Subtract 1 from both sides of the equation.

$$\sqrt{x+15}=x+4-1$$

Subtract 1 from 4 to get 3.

$$\sqrt{x+15}=x+3$$

Square both sides of the equation.

$$\left(\sqrt{x+15}\right)^{2}=\left(x+3\right)^{2}$$

Calculate \sqrt{x+15} to the power of 2 and get x+15.

$$x+15=\left(x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x+15=x^{2}+6x+9$$

Subtract x^{2} from both sides.

$$x+15-x^{2}=6x+9$$

Subtract 6x from both sides.

$$x+15-x^{2}-6x=9$$

Combine x and -6x to get -5x.

$$-5x+15-x^{2}=9$$

Subtract 9 from both sides.

$$-5x+15-x^{2}-9=0$$

Subtract 9 from 15 to get 6.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-6=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=1 b=-6$$

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

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

Factor out x in the first and 6 in the second group.

$$x\left(-x+1\right)+6\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(x+6\right)$$

To find equation solutions, solve -x+1=0 and x+6=0.

$$x=1 x=-6$$

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

$$\sqrt{1+15}+1=1+4$$

Simplify. The value x=1 satisfies the equation.

$$5=5$$

Substitute -6 for x in the equation \sqrt{x+15}+1=x+4.

$$\sqrt{-6+15}+1=-6+4$$

Simplify. The value x=-6 does not satisfy the equation because the left and the right hand side have opposite signs.

$$4=-2$$

Equation \sqrt{x+15}=x+3 has a unique solution.

$$x=1$$