Solve sqrt{x}(5+3sqrt{x})=2

$$\sqrt { x } ( 5 + 3 \sqrt { x } ) = 2$$

Solve for x

$$x=\frac{1}{9}\approx 0.111111111$$

Solution Steps

Use the distributive property to multiply \sqrt{x} by 5+3\sqrt{x}.

$$5\sqrt{x}+3\left(\sqrt{x}\right)^{2}=2$$

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

$$5\sqrt{x}+3x=2$$

Subtract 3x from both sides of the equation.

$$5\sqrt{x}=2-3x$$

Square both sides of the equation.

$$\left(5\sqrt{x}\right)^{2}=\left(2-3x\right)^{2}$$

Expand \left(5\sqrt{x}\right)^{2}.

$$5^{2}\left(\sqrt{x}\right)^{2}=\left(2-3x\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25\left(\sqrt{x}\right)^{2}=\left(2-3x\right)^{2}$$

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

$$25x=\left(2-3x\right)^{2}$$

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

$$25x=4-12x+9x^{2}$$

Add 12x to both sides.

$$25x+12x=4+9x^{2}$$

Combine 25x and 12x to get 37x.

$$37x=4+9x^{2}$$

Subtract 9x^{2} from both sides.

$$37x-9x^{2}=4$$

Subtract 4 from both sides.

$$37x-9x^{2}-4=0$$

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

$$-9x^{2}+37x-4=0$$

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

$$a+b=37 ab=-9\left(-4\right)=36$$

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 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 37.

$$a=36 b=1$$

Rewrite -9x^{2}+37x-4 as \left(-9x^{2}+36x\right)+\left(x-4\right).

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

Factor out 9x in the first and -1 in the second group.

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

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

$$\left(-x+4\right)\left(9x-1\right)$$

To find equation solutions, solve -x+4=0 and 9x-1=0.

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

Substitute 4 for x in the equation \sqrt{x}\left(5+3\sqrt{x}\right)=2.

$$\sqrt{4}\left(5+3\sqrt{4}\right)=2$$

Simplify. The value x=4 does not satisfy the equation.

$$22=2$$

Substitute \frac{1}{9} for x in the equation \sqrt{x}\left(5+3\sqrt{x}\right)=2.

$$\sqrt{\frac{1}{9}}\left(5+3\sqrt{\frac{1}{9}}\right)=2$$

Simplify. The value x=\frac{1}{9} satisfies the equation.

$$2=2$$

Equation 5\sqrt{x}=2-3x has a unique solution.

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