Solve 2sqrt{x}+2x=8+x

$$2 \sqrt{ x } +2x=8+x$$

Solve for x

$$x=4$$

Solution Steps

Subtract 2x from both sides of the equation.

$$2\sqrt{x}=8+x-2x$$

Combine x and -2x to get -x.

$$2\sqrt{x}=8-x$$

Square both sides of the equation.

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

Expand \left(2\sqrt{x}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$4x=64-16x+x^{2}$$

Subtract 64 from both sides.

$$4x-64=-16x+x^{2}$$

Add 16x to both sides.

$$4x-64+16x=x^{2}$$

Combine 4x and 16x to get 20x.

$$20x-64=x^{2}$$

Subtract x^{2} from both sides.

$$20x-64-x^{2}=0$$

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

$$-x^{2}+20x-64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-64. To find a and b, set up a system to be solved.

$$a+b=20 ab=-\left(-64\right)=64$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 64.

$$1,64 2,32 4,16 8,8$$

Calculate the sum for each pair.

$$1+64=65 2+32=34 4+16=20 8+8=16$$

The solution is the pair that gives sum 20.

$$a=16 b=4$$

Rewrite -x^{2}+20x-64 as \left(-x^{2}+16x\right)+\left(4x-64\right).

$$\left(-x^{2}+16x\right)+\left(4x-64\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-16\right)+4\left(x-16\right)$$

Factor out common term x-16 by using distributive property.

$$\left(x-16\right)\left(-x+4\right)$$

To find equation solutions, solve x-16=0 and -x+4=0.

$$x=16 x=4$$

Substitute 16 for x in the equation 2\sqrt{x}+2x=8+x.

$$2\sqrt{16}+2\times 16=8+16$$

Simplify. The value x=16 does not satisfy the equation.

$$40=24$$

Substitute 4 for x in the equation 2\sqrt{x}+2x=8+x.

$$2\sqrt{4}+2\times 4=8+4$$

Simplify. The value x=4 satisfies the equation.

$$12=12$$

Equation 2\sqrt{x}=8-x has a unique solution.

$$x=4$$