Solve sqrt{3x+7}-x=-1

$$\sqrt { 3 x + 7 } - x = - 1$$

Solve for x

$$x=6$$

Solution Steps

Subtract -x from both sides of the equation.

$$\sqrt{3x+7}=-1+x$$

Square both sides of the equation.

$$\left(\sqrt{3x+7}\right)^{2}=\left(-1+x\right)^{2}$$

Calculate \sqrt{3x+7} to the power of 2 and get 3x+7.

$$3x+7=\left(-1+x\right)^{2}$$

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

$$3x+7=1-2x+x^{2}$$

Subtract 1 from both sides.

$$3x+7-1=-2x+x^{2}$$

Subtract 1 from 7 to get 6.

$$3x+6=-2x+x^{2}$$

Add 2x to both sides.

$$3x+6+2x=x^{2}$$

Combine 3x and 2x to get 5x.

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

Subtract x^{2} from both sides.

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

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 positive, the positive number has greater absolute value than the negative. 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.

$$a=6 b=-1$$

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

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

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

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

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

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

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

$$x=6 x=-1$$

Substitute 6 for x in the equation \sqrt{3x+7}-x=-1.

$$\sqrt{3\times 6+7}-6=-1$$

Simplify. The value x=6 satisfies the equation.

$$-1=-1$$

Substitute -1 for x in the equation \sqrt{3x+7}-x=-1.

$$\sqrt{3\left(-1\right)+7}-\left(-1\right)=-1$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-1$$

Equation \sqrt{3x+7}=x-1 has a unique solution.

$$x=6$$