Solve sqrt{4x+8}=x-1

$$\sqrt { 4 x + 8 } = x - 1$$

Solve for x

$$x=7$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4x+8}\right)^{2}=\left(x-1\right)^{2}$$

Calculate \sqrt{4x+8} to the power of 2 and get 4x+8.

$$4x+8=\left(x-1\right)^{2}$$

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

$$4x+8=x^{2}-2x+1$$

Subtract x^{2} from both sides.

$$4x+8-x^{2}=-2x+1$$

Add 2x to both sides.

$$4x+8-x^{2}+2x=1$$

Combine 4x and 2x to get 6x.

$$6x+8-x^{2}=1$$

Subtract 1 from both sides.

$$6x+8-x^{2}-1=0$$

Subtract 1 from 8 to get 7.

$$6x+7-x^{2}=0$$

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

$$-x^{2}+6x+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=6 ab=-7=-7$$

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. The only such pair is the system solution.

$$a=7 b=-1$$

Rewrite -x^{2}+6x+7 as \left(-x^{2}+7x\right)+\left(-x+7\right).

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

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

$$-x\left(x-7\right)-\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x-1\right)$$

To find equation solutions, solve x-7=0 and -x-1=0.

$$x=7 x=-1$$

Substitute 7 for x in the equation \sqrt{4x+8}=x-1.

$$\sqrt{4\times 7+8}=7-1$$

Simplify. The value x=7 satisfies the equation.

$$6=6$$

Substitute -1 for x in the equation \sqrt{4x+8}=x-1.

$$\sqrt{4\left(-1\right)+8}=-1-1$$

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

$$2=-2$$

Equation \sqrt{4x+8}=x-1 has a unique solution.

$$x=7$$