Solve: 8(x-√3)(√3+x)=2x² | Square Root and Quadratic Equation

To solve the equation $8(x-\sqrt{3})(\sqrt{3}+x) = 2x^2$, follow these steps:

• Step 1: Simplify the left side using the difference of squares, which states $(a-b)(a+b)=a^2-b^2$. Here, we treat $a = x$ and $b = \sqrt{3}$. This gives us:

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

• Step 2: Multiply this by 8 as the equation is $8(x^2 - 3)$.

This results in: $8(x^2 - 3) = 8x^2 - 24$

• Step 3: Substitute this into the equation to get: $8x^2 - 24 = 2x^2$.
• Step 4: Rearrange terms and solve for $x$ by bringing all terms to one side:

$8x^2 - 2x^2 - 24 = 0$ which simplifies to $6x^2 - 24 = 0$.

• Step 5: Factor out the common terms in the quadratic equation:

$6(x^2 - 4) = 0$

• Step 6: Notice that using the difference of squares again, $x^2 - 4$ factors to $(x-2)(x+2)$.

This yields: $6(x-2)(x+2) = 0$.

• Step 7: Solve $(x - 2)(x + 2) = 0$ and apply the zero product property:

This gives the solutions $x - 2 = 0$ and $x + 2 = 0$, thus $x = 2$ and $x = -2$.

Therefore, the solution to the problem is $\pm2$.