Difference of squares: Identify the greater value

To solve these equations, we'll follow these steps:

• Step 1: Solve $(\sqrt{x}+4)(\sqrt{x}-4)=0$
• Step 2: Solve $\sqrt{x}=4$
• Step 3: Compare both solutions to find the largest value of $x$

Let's work through each step:

Step 1: For the equation $(\sqrt{x}+4)(\sqrt{x}-4)=0$, note that it is in the form of a difference of squares: $a^2 - b^2 = (a-b)(a+b) = 0$. Here, $a = \sqrt{x}$ and $b = 4$, giving:

$a^2 - b^2 = (\sqrt{x})^2 - 4^2 = x - 16 = 0$

This simplifies to $x - 16 = 0$, meaning $x = 16$.

Step 2: For the equation $\sqrt{x} = 4$, squaring both sides yields:

$(\sqrt{x})^2 = 4^2$ $x = 16$

Step 3: Now that both equations result in $x = 16$, there is only one unique solution. Comparing the single value from each equation, $x = 16$ is the largest.

Therefore, the largest $x$ is $x = 16$.

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 $x = 64$.

Let's solve each equation step-by-step and find the largest $x$.

Equation 1: $3 + x + 17 - 56x = 18x + 44$

This is a linear equation. We will simplify and solve for $x$.

• Combine like terms on the left side: $3 + 17 + x - 56x = 20 - 55x$.
• Equate the simplified expression to the right side: $20 - 55x = 18x + 44$.
• Move all terms involving $x$ to one side: $20 - 44 = 18x + 55x$.
• This simplifies to $-24 = 73x$.
• Solve for $x$ by dividing both sides by 73: $x = -\frac{24}{73}$.

The solution for $x$ from the first equation is $x = -\frac{24}{73}$.

Equation 2: $(x-2)(x+2) = 0$

This equation is a difference of squares, $(x^2 - 4) = 0$.

• Apply the zero-product property: $x - 2 = 0$ or $x + 2 = 0$.
• Solve each equation: $x = 2$ and $x = -2$.

The solutions from the second equation are $x = 2$ and $x = -2$.

Conclusion

From both equations, we have the possible values of $x$ as $-\frac{24}{73}$, $-2$, and $2$.

The largest value among these is $2$.

Therefore, the largest $x$ is $2$.