Difference of squares: Using multiple rules

To solve the equation $(x+3)(x-3) + (x+1)(x-1) = 0$, we will employ the difference of squares formula.

Step 1: Simplify $(x+3)(x-3)$ using the difference of squares:
$(x+3)(x-3) = x^2 - 3^2 = x^2 - 9$.

Step 2: Simplify $(x+1)(x-1)$ using the difference of squares:
$(x+1)(x-1) = x^2 - 1^2 = x^2 - 1$.

Step 3: Substitute the simplified expressions back into the original equation:
$x^2 - 9 + x^2 - 1 = 0$.

Step 4: Combine like terms:
$2x^2 - 10 = 0$.

Step 5: Simplify the equation by factoring or isolating $x^2$:
Divide through by 2 to get $x^2 - 5 = 0$.

Step 6: Solve for $x^2$:
$x^2 = 5$.

Step 7: Solve for $x$ by taking the square root of both sides:
$x = \pm \sqrt{5}$.

Therefore, the solution to the equation is $x = \pm \sqrt{5}$.

First, let's examine the given equation: $x^4 - 16 = (x-2)(x-2)(2+x)(2+x)$.

The expression on the right-hand side can be rewritten and simplified by recognizing it as a repeated multiplication form:

$(x-2)^2(2+x)^2 = [(x-2)(2+x)]^2$.

Now, apply the difference of squares to the product $(x-2)(2+x)$:

$(x-2)(2+x) = x^2 - 4$.

Now, squaring $x^2 - 4$ gives:

$(x^2 - 4)^2 = x^4 - 8x^2 + 16$.

Compare this to the left side of the equation $x^4 - 16$.

The terms do not match, indicating $x^4 - 16$ does not equal $x^4 - 8x^2 + 16$. Thus, originally factoring may contain inherent assumption errors or unrealistic roots if said polynomial equation was proposed as true for all $x$.

However, for the equality $x^4 - 16 = 0$ specifically, solving this separately would involve:

Recognizing the equation as a difference of squares:

$(x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4)$.

Factor further: $(x^2 - 2^2) = (x-2)(x+2)$.

The $x^2 + 4$ does not factor into real numbers as it results in imaginary roots, thus focus remains on the two real roots of the quadratic factor:

$x = \pm 2$.

Therefore, the valid real solutions are found to be $\pm 2$.

Thus, the final correct solution is: $\pm 2$.

To solve this problem, let's follow these steps:

• Step 1: Factor the left-hand side $x^4 - 16$ using the difference of squares.
• Step 2: Expand the right-hand side expression.
• Step 3: Verify if both sides are equivalent after simplification.

Now, let's evaluate each step:

Step 1:
The expression on the left-hand side is $x^4 - 16$.
This can be viewed as $(x^2)^2 - 4^2$,
which is a difference of squares. Therefore, we can factor it as:
$(x^2 - 4)(x^2 + 4)$.

Step 2:
Next, consider the right-hand side, $(x^2 - 4)(4 + x^2)$.
To expand, use distribution (FOIL method):
- First: $x^2 \times 4 = 4x^2$
- Outer: $x^2 \times x^2 = x^4$
- Inner: $-4 \times 4 = -16$
- Last: $-4 \times x^2 = -4x^2$
Combine these terms:
$x^4 + 4x^2 - 16 - 4x^2 = x^4 - 16$.

Step 3:
The expanded term, $x^4 - 16$, matches the factored left-hand side expression, $(x^2-4)(x^2+4)$, showing that both sides are equivalent.

Therefore, the equation $x^4 - 16 = (x^2 - 4)(4 + x^2)$ is True for all values of $x$.

To assess whether the equation $(x+5)(x+3) = x^2 - 3^2$ is a true or false statement, we follow these steps:

• Step 1: Expand the left side using distributive property (FOIL).
• Step 2: Simplify the expression.
• Step 3: Simplify the right side.
• Step 4: Compare both sides of the equation.

Step 1: Expand $(x+5)(x+3)$:
$(x+5)(x+3) = x(x+3) + 5(x+3)$
$= x^2 + 3x + 5x + 15$

Step 2: Simplify this expression:
$x^2 + 3x + 5x + 15 = x^2 + 8x + 15$

Step 3: Evaluate the right side:
$x^2 - 3^2 = x^2 - 9$

Step 4: Compare $x^2 + 8x + 15$ to $x^2 - 9$:
Clearly, $x^2 + 8x + 15 \neq x^2 - 9$, as the former includes the terms $8x + 15$, while the latter is simply reduced by 9.

Therefore, the equation $(x+5)(x+3) = x^2 - 3^2$ is a Lie (false statement) because the left and right sides do not match for any value of $x$.

To solve this problem, we'll follow these steps:

• Step 1: Expand both sides of the given equation.
• Step 2: Compare and simplify the resulting expressions.
• Step 3: Solve for $a$ and $b$.

Now, let's work through each step:
Step 1: The given equation is $(a+b)(a-b) = (a+b)^2$. Let's expand both sides:
- Left side: $(a+b)(a-b) = a^2 - b^2$ based on the difference of squares formula.
- Right side: $(a+b)^2 = a^2 + 2ab + b^2$ using the square of a sum formula.

Step 2: Setting the expanded forms equal gives us:
$a^2 - b^2 = a^2 + 2ab + b^2$.

Step 3: Simplify and solve the equation:
- Subtract $a^2$ from both sides: $-b^2 = 2ab + b^2$.
- Add $b^2$ to both sides: $0 = 2ab + 2b^2$.
- Factor the right-hand side: $0 = 2b(a + b)$.

This gives us two possible conditions:
1) $2b = 0$, which implies $b = 0$.
2) $a + b = 0$, which implies $a = -b$.

Since $a = -b$ satisfies the equation for any $a$ if $b$ is not zero, and when $b = 0$, the equation simplifies to $0 = 0$, both conditions are valid.

Therefore, the solutions are $a = -b$ or $b = 0$.

In conclusion, the answer is: $a=-b$ or $0=b$.

To solve the problem, we will follow these steps:

• Step 1: Expand the square of the binomial $(3y + 4a)$.
• Step 2: Simplify the product of the binomials $9(y-2a)(y+2a)$.
• Step 3: Subtract the second result from the first and simplify.

Let's work through each step:
Step 1: The expression $(3y + 4a)^2$ can be expanded as follows:

$(3y + 4a)^2 = (3y)^2 + 2(3y)(4a) + (4a)^2 = 9y^2 + 24ay + 16a^2$.

Step 2: Simplify the expression $9(y-2a)(y+2a)$. This uses the difference of squares formula where $(y-2a)(y+2a) = y^2 - (2a)^2$:

$(y-2a)(y+2a) = y^2 - 4a^2$.

Then multiply by 9:

$9(y^2 - 4a^2) = 9y^2 - 36a^2$.

Step 3: Now subtract the second result from the first:

$(9y^2 + 24ay + 16a^2) - (9y^2 - 36a^2) = 9y^2 + 24ay + 16a^2 - 9y^2 + 36a^2$.

Combine and simplify:

$24ay + 52a^2$.

Factoring out $4a$ from the expression:

$4a(6y + 13a)$.

Therefore, the solution to the problem is: $4a(6y + 13a)$.

Let's solve this problem by simplifying each component separately:

First, simplify $(\frac{x}{3} - 4)^2$ using the square of a difference formula:

$(\frac{x}{3} - 4)^2 = \left(\frac{x}{3}\right)^2 - 2 \times \frac{x}{3} \times 4 + 4^2$ This becomes:

$= \frac{x^2}{9} - \frac{8x}{3} + 16$

Next, simplify $x(\frac{\sqrt{8x}}{3} + 2)(\frac{\sqrt{8x}}{3} - 2)$ using the difference of squares formula:

$(\frac{\sqrt{8x}}{3} + 2)(\frac{\sqrt{8x}}{3} - 2) = \left(\frac{\sqrt{8x}}{3}\right)^2 - 2^2$ Simplify further:

$= \frac{8x}{9} - 4$

Including the factor of $x$, we have:

$x \left(\frac{8x}{9} - 4\right) = \frac{8x^2}{9} - 4x$

Combine the results from both parts:

$\left(\frac{x^2}{9} - \frac{8x}{3} + 16\right) + \left(\frac{8x^2}{9} - 4x\right)$

Simplify by combining like terms:

$= \frac{x^2}{9} + \frac{8x^2}{9} - \frac{8x}{3} - 4x + 16 = x^2 - \left(\frac{8x}{3} + 4x\right) + 16 = x^2 - \left(\frac{8x + 12x}{3}\right) + 16 = x^2 - \frac{20x}{3} + 16$

Therefore, after simplifying, the expression becomes $\boldsymbol{x^2 - \frac{20x}{3} + 16}$.

The final solution is: $x^2 - \frac{20x}{3} + 16$.

To solve this problem, let's follow a detailed approach:

• First, expand both squares:
• $\left(\frac{2}{3} + \frac{m}{4}\right)^2 = \left(\frac{2}{3}\right)^2 + \frac{2 \times 2}{3 \times 4} \times m + \left(\frac{m}{4}\right)^2 = \frac{4}{9} + \frac{m}{6} + \frac{m^2}{16}$
• $\left(\frac{m}{4} - \frac{2}{3}\right)^2 = \left(\frac{m}{4}\right)^2 - \frac{2 \times m}{12} + \left(\frac{2}{3}\right)^2 = \frac{m^2}{16} - \frac{m}{6} + \frac{4}{9}$
• Now substitute and simplify into the expression:

• Expression becomes: $\left( \frac{4}{9} + \frac{m}{6} + \frac{m^2}{16} \right) - \frac{4}{3} - \left( \frac{m^2}{16} - \frac{m}{6} + \frac{4}{9} \right)$

• Observe that $\frac{m^2}{16} - \frac{m^2}{16}$ cancels out.
• Now simplify the remaining terms: $\frac{4}{9} + \frac{m}{6} - \frac{4}{3} + \frac{m}{6} - \frac{4}{9} = \frac{2m}{6} - \frac{4}{3}$
• Use common denominators to combine final terms:
• $\frac{2m}{6} = \frac{m}{3}$, $\frac{4}{3} = \frac{12}{9}$, resulting in: $\frac{m}{3} - \frac{12}{9}$
• Recognize that this is a difference of squares:
• $\frac{m}{3} - \frac{12}{9} = \frac{(\sqrt{2m} + 2)(\sqrt{2m} - 2)}{3}$

Therefore, the simplified expression is given by the choice: $\frac{(\sqrt{2m}+2)(\sqrt{2m}-2)}{3}$.