Square of Difference: System of equations with no solution

We use the abbreviated multiplication formula:

$(x-y)(x-y)=$

$x^2-xy-yx+y^2=$

$x^2-2xy+y^2$

To solve the problem, we need to expand the expression $(x-7)^2$ using the formula for the square of a difference.

The formula for the square of a difference is $(a-b)^2 = a^2 - 2ab + b^2$.

Let's apply this formula to our expression $(x-7)^2$:

• Identify $a = x$ and $b = 7$.
• Substitute these values into the formula: $(x-7)^2 = x^2 - 2(x)(7) + 7^2$.
• Calculate each term:
• $x^2$ remains as $x^2$.
• $-2(x)(7) = -14x$.
• $7^2 = 49$.

So, expanding the expression, we get $x^2 - 14x + 49$.

Thus, the expression that has the same value as $(x-7)^2$ is $x^2 - 14x + 49$.

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

• Step 1: Identify a and b in the expression $(x^2 - 6)^2$.
• Step 2: Apply the square of a difference formula.
• Step 3: Simplify the resulting expression.

Now, let's work through each step:
Step 1: The expression is $(x^2 - 6)^2$. Here, $a = x^2$ and $b = 6$.
Step 2: Apply the binomial formula: $(a - b)^2 = a^2 - 2ab + b^2$.
Step 3:
1. Calculate $a^2$:
$a^2 = (x^2)^2 = x^4$.
2. Calculate $2ab$:
$2ab = 2(x^2)(6) = 12x^2$.
3. Calculate $b^2$:
$b^2 = 6^2 = 36$.
4. Substitute these back into the formula:
$(x^2 - 6)^2 = x^4 - 12x^2 + 36$.

Therefore, the expanded expression is $x^4 - 12x^2 + 36$.

To solve the expression $(x-x^2)^2$, we will use the square of a binomial formula $(a-b)^2 = a^2 - 2ab + b^2$.

Let's identify $a$ and $b$ in our expression:

• Here, $a = x$ and $b = x^2$.

Applying the formula:

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

Calculating each part, we get:

• $(x)^2 = x^2$
• $-2(x)(x^2) = -2x^3$
• $(x^2)^2 = x^4$

Combining these results, the expression simplifies to:

$x^4 - 2x^3 + x^2$

Therefore, the expanded form of $(x-x^2)^2$ is $\boxed{x^4 - 2x^3 + x^2}$.

To rewrite the expression $9x^2 - 12x + 4$ as a perfect square, follow these steps:

• Step 1: Compare the expression $9x^2 - 12x + 4$ with $(a - b)^2$.
• Step 2: Note that $(a - b)^2 = a^2 - 2ab + b^2$.
• Step 3: Identify matching terms: $a^2 = 9x^2$, $-2ab = -12x$, $b^2 = 4$.

Now, separate this into steps:
Step 1: Set $a^2 = 9x^2$, so $a = 3x$.
Step 2: Set $b^2 = 4$, so $b = 2$.
Step 3: Verify $-2ab = -12x$:
$-2 \times 3x \times 2 = -12x.$
This confirms our values of $a$ and $b$ are correct.

Thus, the expression $9x^2 - 12x + 4$ is equivalent to the square $(3x - 2)^2$.

The correct choice is: $(3x-2)^2$.

In order to solve the question, we need to know one of the shortcut multiplication formulas:

$(x−y)^2=x^2−2xy+y^2$

We apply the formula twice:

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

$(x-3)^2=x^2-6x+9$

Now we add the two together:

$x^2-4x+4+x^2-6x+9=$

$2 x^2-10x+13$

To solve the expression $2(3x-1)^2 - 3(2x+1)^2$, we perform these steps:

• First, expand and simplify $(3x-1)^2$:
  $(3x-1)^2 = (3x)^2 - 2(3x)(1) + 1^2 = 9x^2 - 6x + 1$.
• Next, expand and simplify $(2x+1)^2$:
  $(2x+1)^2 = (2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1$.
• Now, multiply these by their corresponding coefficients and subtract:
  $2(3x-1)^2 = 2(9x^2 - 6x + 1) = 18x^2 - 12x + 2$,
  $3(2x+1)^2 = 3(4x^2 + 4x + 1) = 12x^2 + 12x + 3$.
• Subtract the second expression from the first:
  $18x^2 - 12x + 2 - (12x^2 + 12x + 3) = 18x^2 - 12x + 2 - 12x^2 - 12x - 3$.
• Simplify the expression:
  $6x^2 - 24x - 1$ is obtained by combining like terms.
• The final expression simplifies to $6x(x-4) - 1$ by factoring out $6x$.

The correct answer is $\mathbf{6x(x-4)-1}$, which corresponds to choice 3.