Square of sum: Data with powers and roots

To solve the problem, we undertake the following steps:

• Identify the components $a = x$ and $b = \sqrt{x}$ in the expression $(x + \sqrt{x})^2$.
• Apply the square of sum formula: $(a + b)^2 = a^2 + 2ab + b^2$.
• Substitute the values:
  • $a^2 = x^2$
  • $2ab = 2 \times x \times \sqrt{x} = 2x\sqrt{x}$
  • $b^2 = (\sqrt{x})^2 = x$
• Simplifying gives: $x^2 + 2x\sqrt{x} + x$
• Recognize that this can be factored to yield: $x(x + 2\sqrt{x} + 1)$

Thus, the simplified expression is $\boxed{x[x + 2\sqrt{x} + 1]}$.

In order to solve the equation, start by removing the denominators.

To do this, we'll multiply the denominators:

$(2x+1)^2\cdot(2x+1)+(x+2)^2\cdot(x+2)=4.5x(2x+1)(x+2)$

Open the parentheses on the left side, making use of the distributive property:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=4.5x(2x+1)(x+2)$

Continue to open the parentheses on the right side of the equation:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=4.5x(2x^2+5x+2)$

Simplify further:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=9x^3+22.5x+9x$

Go back and simplify the parentheses on the left side of the equation:

$8x^3+8x^2+2x+4x^2+4x+1+x^3+4x^2+4x+2x^2+8x+8=9x^3+22.5x+9x$

Combine like terms:

$9x^3+18x^2+18x+9=9x^3+22.5x+9x$

Notice that all terms can be divided by 9 as shown below:

$x^3+2x^2+2x+1=x^3+2.5x+x$

Move all numbers to one side:

$x^3-x^3+2x^2-2.5x^2+2x-x+9=0$

We obtain the following:

$0.5x^2-x-1=0$

In order to remove the one-half coefficient, multiply the entire equation by 2

$x^2-2x-2=0$

Apply the square root formula, as shown below-

$\frac{2±\sqrt{12}}{2}$

Apply the properties of square roots in order to simplify the square root of 12:

$\frac{2±2\sqrt{3}}{2}$Divide both the numerator and denominator by 2 as follows:

$1±\sqrt{3}$