Solve sqrt{3x+28}-8=x

$$\sqrt { 3 x + 28 } - 8 = x$$

Solve for x

$$x=-4$$

Solution Steps

Subtract -8 from both sides of the equation.

$$\sqrt{3x+28}=x+8$$

Square both sides of the equation.

$$\left(\sqrt{3x+28}\right)^{2}=\left(x+8\right)^{2}$$

Calculate \sqrt{3x+28} to the power of 2 and get 3x+28.

$$3x+28=\left(x+8\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+8\right)^{2}.

$$3x+28=x^{2}+16x+64$$

Subtract x^{2} from both sides.

$$3x+28-x^{2}=16x+64$$

Subtract 16x from both sides.

$$3x+28-x^{2}-16x=64$$

Combine 3x and -16x to get -13x.

$$-13x+28-x^{2}=64$$

Subtract 64 from both sides.

$$-13x+28-x^{2}-64=0$$

Subtract 64 from 28 to get -36.

$$-13x-36-x^{2}=0$$

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

$$-x^{2}-13x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=-13 ab=-\left(-36\right)=36$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -13.

$$a=-4 b=-9$$

Rewrite -x^{2}-13x-36 as \left(-x^{2}-4x\right)+\left(-9x-36\right).

$$\left(-x^{2}-4x\right)+\left(-9x-36\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(-x-4\right)+9\left(-x-4\right)$$

Factor out common term -x-4 by using distributive property.

$$\left(-x-4\right)\left(x+9\right)$$

To find equation solutions, solve -x-4=0 and x+9=0.

$$x=-4 x=-9$$

Substitute -4 for x in the equation \sqrt{3x+28}-8=x.

$$\sqrt{3\left(-4\right)+28}-8=-4$$

Simplify. The value x=-4 satisfies the equation.

$$-4=-4$$

Substitute -9 for x in the equation \sqrt{3x+28}-8=x.

$$\sqrt{3\left(-9\right)+28}-8=-9$$

Simplify. The value x=-9 does not satisfy the equation.

$$-7=-9$$

Equation \sqrt{3x+28}=x+8 has a unique solution.

$$x=-4$$