Solve 2x=2sqrt{2x}+3

$$2 x = 2 \sqrt { 2 x } + 3$$

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Solution Steps

Subtract 3 from both sides of the equation.

$$2x-3=2\sqrt{2x}$$

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.

$$4x^{2}-12x+9=\left(2\sqrt{2x}\right)^{2}$$

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

$$4x^{2}-12x+9=2^{2}\left(\sqrt{2x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-12x+9=4\left(\sqrt{2x}\right)^{2}$$

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

$$4x^{2}-12x+9=4\times 2x$$

Multiply 4 and 2 to get 8.

$$4x^{2}-12x+9=8x$$

Subtract 8x from both sides.

$$4x^{2}-12x+9-8x=0$$

Combine -12x and -8x to get -20x.

$$4x^{2}-20x+9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-20 ab=4\times 9=36$$

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 -20.

$$a=-18 b=-2$$

Rewrite 4x^{2}-20x+9 as \left(4x^{2}-18x\right)+\left(-2x+9\right).

$$\left(4x^{2}-18x\right)+\left(-2x+9\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(2x-9\right)-\left(2x-9\right)$$

Factor out common term 2x-9 by using distributive property.

$$\left(2x-9\right)\left(2x-1\right)$$

To find equation solutions, solve 2x-9=0 and 2x-1=0.

$$x=\frac{9}{2} x=\frac{1}{2}$$

Substitute \frac{9}{2} for x in the equation 2x=2\sqrt{2x}+3.

$$2\times \frac{9}{2}=2\sqrt{2\times \frac{9}{2}}+3$$

Simplify. The value x=\frac{9}{2} satisfies the equation.

$$9=9$$

Substitute \frac{1}{2} for x in the equation 2x=2\sqrt{2x}+3.

$$2\times \frac{1}{2}=2\sqrt{2\times \frac{1}{2}}+3$$

Simplify. The value x=\frac{1}{2} does not satisfy the equation.

$$1=5$$

Equation 2x-3=2\sqrt{2x} has a unique solution.

$$x=\frac{9}{2}$$