Solve x+sqrt{5x+19}=-1

$$x+ \sqrt{ 5x+19 } =-1$$

Solve for x

$$x=-3$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{5x+19}=-1-x$$

Square both sides of the equation.

$$\left(\sqrt{5x+19}\right)^{2}=\left(-1-x\right)^{2}$$

Calculate \sqrt{5x+19} to the power of 2 and get 5x+19.

$$5x+19=\left(-1-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-1-x\right)^{2}.

$$5x+19=1+2x+x^{2}$$

Subtract 1 from both sides.

$$5x+19-1=2x+x^{2}$$

Subtract 1 from 19 to get 18.

$$5x+18=2x+x^{2}$$

Subtract 2x from both sides.

$$5x+18-2x=x^{2}$$

Combine 5x and -2x to get 3x.

$$3x+18=x^{2}$$

Subtract x^{2} from both sides.

$$3x+18-x^{2}=0$$

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

$$-x^{2}+3x+18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=3 ab=-18=-18$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 3.

$$a=6 b=-3$$

Rewrite -x^{2}+3x+18 as \left(-x^{2}+6x\right)+\left(-3x+18\right).

$$\left(-x^{2}+6x\right)+\left(-3x+18\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-6\right)-3\left(x-6\right)$$

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

$$\left(x-6\right)\left(-x-3\right)$$

To find equation solutions, solve x-6=0 and -x-3=0.

$$x=6 x=-3$$

Substitute 6 for x in the equation x+\sqrt{5x+19}=-1.

$$6+\sqrt{5\times 6+19}=-1$$

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

$$13=-1$$

Substitute -3 for x in the equation x+\sqrt{5x+19}=-1.

$$-3+\sqrt{5\left(-3\right)+19}=-1$$

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

$$-1=-1$$

Equation \sqrt{5x+19}=-x-1 has a unique solution.

$$x=-3$$