Square of Difference - Examples, Exercises and Solutions

To solve this problem, we will apply the square of a binomial formula.

The given expression is $(4b-3)(4b-3)$. We recognize this as the square of a binomial, which can be rewritten as $(4b-3)^2$. To expand this expression, we use the formula:

$(a-b)^2 = a^2 - 2ab + b^2$

In our expression, $a = 4b$ and $b = 3$. Let's apply the formula:

• Calculate $a^2$:
  $a^2 = (4b)^2 = 16b^2$
• Calculate $-2ab$:
  $-2ab = -2(4b)(3) = -24b$
• Calculate $b^2$:
  $b^2 = (3)^2 = 9$

Putting it all together, we have:

$(4b-3)^2 = 16b^2 - 24b + 9$

Therefore, the exponential summation expression is $(4b-3)^2$, with the expanded form:

$16b^2 - 24b + 9$

This matches choice 3, confirming our solution.

To solve this problem, let's start by identifying the parts of the binomial:

• The expression $(3x-y)^2$ represents a binomial squared.
• We recognize it has the form $(a-b)^2$ where $a = 3x$ and $b = y$.
• Using the formula for the square of a difference: $(a-b)^2 = a^2 - 2ab + b^2$, we find the expanded form.

Let's apply the formula:

Step 1: Expand $(3x-y)^2$ using the formula:
$(3x-y)^2 = (3x)^2 - 2(3x)(y) + y^2$

Step 2: Calculate each part:
$(3x)^2 = 9x^2$
$-2(3x)(y) = -6xy$
$y^2$ stays as $y^2$

Step 3: Combine these results to get the addition form:
$9x^2 - 6xy + y^2$

The expression in multiplication form, as provided, is just repeating the factors:
$(3x-y)(3x-y)$

Therefore, the expression rewritten as addition is $9x^2 - 6xy + y^2$ and as multiplication $(3x-y)(3x-y)$.

To solve the problem, we will expand the expression $(a-4)(a-4)$ using the square of a difference formula.

This formula states: $(x-y)^2 = x^2 - 2xy + y^2$.
In our case, $x = a$ and $y = 4$, so we apply the formula:

• First term: $x^2 = a^2$
• Second term: $-2xy = -2 \cdot a \cdot 4 = -8a$
• Third term: $y^2 = 4^2 = 16$

Putting it all together, the expression becomes:
$a^2 - 8a + 16$.

After matching this result with the given choices, we find it corresponds to choice 4.

Therefore, the solution to the problem is $\mathbf{a^2 - 8a + 16}$.

To solve for $(7b - 3x)^2$ as a sum, we'll follow these steps:

• Step 1: Identify the given expression and apply the formula for the square of a difference:
  $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 7b$ and $b = 3x$.
• Step 2: Expand each term:
• $a^2 = (7b)^2 = 49b^2$
• $-2ab = -2 \times 7b \times 3x = -42bx$
• $b^2 = (3x)^2 = 9x^2$
• Step 3: Combine all terms to form the sum:
  $(7b - 3x)^2 = 49b^2 - 42bx + 9x^2$.

Therefore, the solution to the problem is $(7b - 3x)^2 = 49b^2 - 42bx + 9x^2$.

Hence, the correct answer choice is: $49b^2 - 42bx + 9x^2$

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: $(3x-2)^2$.

We will solve the quadratic equation $x^2 - 2x + 1 = 9$ using the square of a binomial formula.

Firstly, let's recognize that the left side of the equation forms a perfect square:

$x^2 - 2x + 1 \equiv (x - 1)^2$

Therefore, the equation can be rewritten as:

$(x - 1)^2 = 9$

To solve for $x$, take the square root of both sides. Remember to consider both the positive and negative solutions from the square root:

Thus, $x - 1 = \pm 3$

This gives us two separate equations to solve:

• $x - 1 = 3$
• $x - 1 = -3$

Solving each equation for $x$ gives:

• For $x - 1 = 3$:

Add 1 to both sides: $x = 4$

• For $x - 1 = -3$:

Add 1 to both sides: $x = -2$

Therefore, the solutions to the equation are $x = 4$ and $x = -2$.

Comparing these solutions to the given answer choices, we identify the correct choice as:

$x=-2$ or $x=4$

In conclusion, the solutions to the equation are $x = 4$ and $x = -2$.

Proceed to solve the given equation:

$x^2+144=24x$

Arrange the equation by moving terms:

$x^2+144=24x \\ x^2-24x+144=0$

Note that we are able to factor the expression on the left side by using the perfect square trinomial formula:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

As demonstrated below:

$144=12^2$

Therefore, we'll represent the rightmost term as a squared term:

$x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-24x+\textcolor{blue}{12}^2=0$

Now let's examine once again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation that we obtained in the last step:

$\textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0$

Note that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{12}^2$indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor this expression (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we'll query whether we can represent the expression on the left side of the equation as:

$\textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0$

And indeed it is true that:

$2\cdot x\cdot12=24x$

Therefore we can represent the expression on the left side of the equation as a perfect square trinomial:

$\textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0$

From here we can take the square root of both sides of the equation (and don't forget that there are two possibilities - positive and negative when taking an even root of both sides of an equation), then we'll easily solve by isolating the variable:

$(x-12)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-12=\pm0\\ x-12=0\\ \boxed{x=12}$

Let's summarize the solution of the equation:

$x^2+144=24x \\ x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0 \\ \downarrow\\ x-12=0\\ \downarrow\\ \boxed{x=12}$

Therefore the correct answer is answer C.

Proceed to solve the given equation:

$x^2=6x-9$

First, let's arrange the equation by moving terms:

$x^2=6x-9 \\ x^2-6x+9=0$

Note that we can factor the expression on the left side by using the perfect square trinomial formula for a binomial squared:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

As shown below:

$9=3^2$Therefore, we'll represent the rightmost term as a squared term:

$x^2-6x+9=0 \\ \downarrow\\ \textcolor{red}{x}^2-6x+\textcolor{blue}{3}^2=0$

Now let's examine again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation that we obtained in the last step:

$\textcolor{red}{x}^2-\underline{6x}+\textcolor{blue}{3}^2=0$

Note that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{3}^2$indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor the expression in question (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined with a single line):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we will query whether we can represent the expression on the left side of the equation as:

$\textcolor{red}{x}^2-\underline{6x}+\textcolor{blue}{3}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}}+\textcolor{blue}{3}^2=0$

And indeed it is true that:

$2\cdot x\cdot3=6x$

Therefore we can represent the expression on the left side of the equation as a perfect square binomial:

$\textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}}+\textcolor{blue}{3}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{3})^2=0$

From here we can take the square root of both sides of the equation (and don't forget that there are two possibilities - positive and negative when taking an even root of both sides of an equation), then we'll easily solve by isolating the variable:

$(x-3)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-3=\pm0\\ x-3=0\\ \boxed{x=3}$

Let's summarize the solution of the equation:

$x^2=6x-9 \\ x^2-6x+9=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}+\textcolor{blue}{3}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{3})^2=0 \\ \downarrow\\ x-3=0\\ \downarrow\\ \boxed{x=3}$

Therefore the correct answer is answer C.

To address this mathematical problem, we will apply the square of a binomial formula and solve for the missing term. Here's how:

• Step 1: Expand the right-hand side of the equation $(x-3)^2$.
• Step 2: Equate the expanded form to $x^2 + ? + 9$.
• Step 3: Solve for the missing term by comparing the coefficients.

Step 1: Expanding $(x-3)^2$ using the formula, we get:

$(x - 3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2$.

This simplifies to:

$x^2 - 6x + 9$.

Step 2: Equating this to the left-hand side:

$x^2 + ? + 9 = x^2 - 6x + 9$.

Step 3: Compare the terms:

The term that replaces "?" on the left-hand side must make the equation hold.

Setting corresponding terms equal, we find that:

$? = -6x$.

Therefore, the solution to the problem is $-6x$.

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

• Step 1: Identify the given form and match it with the square of a difference formula
• Step 2: Determine the values of 'a' and 'b' such that the expanded form matches both sides of the equation
• Step 3: Calculate the missing value in the expression

Now, let's work through each step:
Step 1: We are given the expression $(2x-?)^2=4x^2-12x+\text{?}$.
Step 2: Using the standard formula for a perfect square expansion:
$(2x-b)^2 = (2x)^2 - 2 \cdot 2x \cdot b + b^2 = 4x^2 - 4xb + b^2$.
By matching coefficients, in $4x^2 - 12x + \text{?}$, we see $4xb = 12x$. Thus, $b = 3$.
Step 3: Substitute $b = 3$ into $b^2$ to get the constant term: $b^2 = 3^2 = 9$.

Therefore, the solution to the problem is $3,\text{ }9$.

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

• Step 1: Recognize this as a binomial square problem.
• Step 2: Identify the expanded form of a binomial square.
• Step 3: Match the terms from both sides of the equation.

Now, let's work through each step:
Step 1: The given expression $(x-?)^2 = x^2 - ? + 25$ is the expansion of a binomial $(x-a)^2$.
Step 2: Recall that $(x-a)^2 = x^2 - 2ax + a^2$.
Step 3: Compare this equivalent form to $x^2 - ? + 25$:

- The term $x^2$ matches directly on both sides.
- The constant term $a^2 = 25$, so $a = 5$ because $5^2 = 25$.
- The middle term $-2ax$ is currently unspecified, but it provides the form needed to fill the blank with $-2ax$.

Matching the expanded form: - The term inside $(x-?)$ should be $5$ (because $a=5$).
- Therefore, the missing linear term can be $10x$ since $-2 \times 5 = -10$.

Therefore, the solution to the problem is $5,\text{ }10x$.