Square of Difference: Transition between expressions

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: $49b^2 - 42bx + 9x^2$

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.

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.

Let's solve the given equation:

$60-16y+y^2=-4$First, let's arrange the equation by moving terms:

$60-16y+y^2=-4 \\ 60-16y+y^2+4=0 \\ y^2-16y+64=0$Now, let's note that we can break down the expression on the left side using the short quadratic factoring formula:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$This is done using the fact that:

$64=8^2$So let's present the outer term on the right as a square:

$y^2-16y+64=0 \\ \downarrow\\ \textcolor{red}{y}^2-16y+\textcolor{blue}{8}^2=0$Now let's examine again the short factoring formula we 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 of the equation we got in the last step:

$\textcolor{red}{y}^2-\underline{16y}+\textcolor{blue}{8}^2=0$Let's note that the terms $\textcolor{red}{y}^2,\hspace{6pt}\textcolor{blue}{8}^2$indeed match the form of the first and third terms in the short multiplication formula (which are highlighted in red and blue),

But in order for us to break down the relevant expression (which is on the left side of the equation) using the short formula we mentioned, the match to the short formula must also apply to the remaining term, 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 ask if it's possible to present the expression on the left side of the equation as:

$\textcolor{red}{y}^2-\underline{16y}+\textcolor{blue}{8}^2 =0 \\ \updownarrow\text{?}\\ \textcolor{red}{y}^2-\underline{2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{8}}+\textcolor{blue}{8}^2 =0$And indeed it holds that:

$2\cdot y\cdot8=16y$So we can present the expression on the left side of the given equation as a difference of two squares:

$\textcolor{red}{y}^2-2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{8}+\textcolor{blue}{8}^2=0 \\ \downarrow\\ (\textcolor{red}{y}-\textcolor{blue}{8})^2=0$From here we can take out square roots for the two sides of the equation (remember that there are two possibilities - positive and negative when taking out square roots), we'll solve it easily by isolating the variable on one side:

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

Let's summarize then the solution of the equation:

$60-16y+y^2=-4 \\ y^2-16y+64=0 \\ \downarrow\\ \textcolor{red}{y}^2-2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{8}+\textcolor{blue}{8}^2=0 \\ \downarrow\\ (\textcolor{red}{y}-\textcolor{blue}{8})^2=0 \\ \downarrow\\ y-8=0\\ \downarrow\\ \boxed{y=8}$

So the correct answer is answer a.

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

• Step 1: Simplify the given expression using order of operations.
• Step 2: Match the simplified expression with an algebraic identity to see if it's a square of a difference.

Now, let's work through each step:

Step 1: The given expression is $49 - 2 \times 21 + 9$. According to the order of operations, we will perform the multiplication first:

$2 \times 21 = 42$.

So, the expression becomes:

$49 - 42 + 9$.

Performing the subtraction gives us:

$49 - 42 = 7$.

Addition follows:

$7 + 9 = 16$.

Step 2: Now, we have to express 16 as a square of a difference. We find that:

$16 = (4)^2$.

Given the choices relate to squares of expressions in the form $(a-b)^2$, let's try expressing 16 as a square of the difference of two numbers:

The nearest choice is $(7-3)^2 = 7^2 - 2 \times 7 \times 3 + 3^2 = 49 - 42 + 9 = 16$.

Therefore, the simplified expression can be written as $(7-3)^2$.

Thus, the correct choice is $(7-3)^2$.

To solve this problem, we'll simplify the expression $\frac{m^2}{9} - \frac{4}{3}mn + 4n^2$ to reveal a possible square of a binomial:

• Step 1: Recognize the expression $\frac{m^2}{9} - \frac{4}{3}mn + 4n^2$ may have terms fitting the identity $(a - b)^2 = a^2 - 2ab + b^2$.
• Step 2: Identify parts of the expression:
  - $a^2 = \frac{m^2}{9}$, so $a = \frac{m}{3}$.
  - $-2ab = -\frac{4}{3}mn$. Since $a = \frac{m}{3}$, solve for $b$:
  $2 \times \frac{m}{3} \times b = \frac{4}{3}mn \Rightarrow 2b = 4n \Rightarrow b = 2n$.
• Step 3: Check $b^2$: $b^2 = (2n)^2 = 4n^2$, which matches the $4n^2$ term in the expression.

Therefore, the correct form of the expression is:
$(\frac{m}{3} - 2n)^2$.

Thus, the expression simplifies to:

$(\frac{m}{3} - 2n)^2$

To solve this problem, we will first rewrite the given expression $(mn-l)^2$ using the binomial square formula.

The formula for the square of a binomial is:

• $(a - b)^2 = a^2 - 2ab + b^2$

In the expression $(mn-l)^2$, identify $a = mn$ and $b = l$.

Apply the binomial square formula:

$(mn - l)^2 = (mn)^2 - 2(mn)(l) + l^2$

Calculate each term:

• $(mn)^2 = m^2n^2$
• $2(mn)(l) = 2mnl$
• $l^2 = l^2$

Combine these to write the expression as a sum of terms:

$m^2n^2 - 2mnl + l^2$

Next, express $(mn-l)^2$ as a multiplication:

By definition, the square of a term can be rewritten as the term multiplied by itself:

$(mn-l)^2 = (mn-l)(mn-l)$

Therefore, the expression rewritten as an addition and a multiplication is:

$m^2n^2-2mnl+l^2$

$(mn-l)(mn-l)$