Solve and Compare: Delving into (√x+8)(√x-8)=0 and x²-100=0

To solve each equation separately and find the largest value for $x$, follow these steps:

• Step 1: Solve $(\sqrt{x}+8)(\sqrt{x}-8)=0$.
• Step 2: Solve $x^2-100=0$.
• Step 3: Compare the solutions to determine the largest $x$.

Now, let's solve each step:

**Step 1**: For the equation $(\sqrt{x}+8)(\sqrt{x}-8)=0$, apply the zero-product property:

- $\sqrt{x}+8=0$ or $\sqrt{x}-8=0$.
Solving these, we get:
From $\sqrt{x}+8=0$:
$\sqrt{x} = -8$, which has no real solutions since square roots cannot be negative.
From $\sqrt{x}-8=0$:
$\sqrt{x} = 8$, thus $x = 8^2 = 64$.

**Step 2**: For the equation $x^2-100=0$, recognize it as a difference of squares $x^2-10^2=0$:

- Factoring gives $(x-10)(x+10)=0$.
Solving these, we find:
$x-10=0$, so $x=10$.
$x+10=0$, so $x=-10$.

**Step 3**: Compare the solutions of both equations to determine the largest $x$:

- From Step 1, the valid solution is $x=64$. - From Step 2, the valid solutions are $x=10$ and $x=-10$.
The largest $x$ is $64$ from the solutions obtained.

Therefore, the largest solution for $x$ is $x = 64$.