Square of sum: Transition between expressions

To solve this problem, we'll use the formula for the square of a sum.

• Step 1: Identify our variables as $a = 7$ and $b = 8$.
• Step 2: Apply the formula $(a + b)^2 = a^2 + 2ab + b^2$.
• Step 3: Substitute into the formula:
  $(7 + 8)^2 = 7^2 + 2 \times 7 \times 8 + 8^2$.
• Step 4: Express the final expression in the most informative form.

Therefore, the expanded expression for $(7 + 8)^2$ is $7^2 + 2 \times 7 \times 8 + 8^2$.

Regarding the choices provided, the correct one is Option 3: $7^2 + 2 \times 7 \times 8 + 8^2$.

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

• Identify the given expression $(a+b)^2$.
• Apply the formula for the square of a sum: $(x+y)^2 = x^2 + 2xy + y^2$.
• Substitute $x = a$ and $y = b$ into the formula and simplify.

Let's apply these steps to the expression $(a+b)^2$:
We start with the expression $(a+b)^2$. This means we are squaring the sum $a + b$.

According to the formula $(x+y)^2 = x^2 + 2xy + y^2$, we can substitute $x = a$ and $y = b$. Therefore, the expression becomes:

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

Therefore, the expanded form of the expression $(a+b)^2$ is $a^2 + 2ab + b^2$.

According to the shortened multiplication formula:

Since 7 and X appear twice, we raise both terms to the power:

$(7+x)^2$

To solve the problem $(3x + 4)^2$, we will use the formula for the square of a binomial:

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

• Identify $a$ and $b$: Here, $a = 3x$ and $b = 4$.
• Apply the formula:

1. Calculate $a^2$ which is $(3x)^2 = 9x^2$.

2. Calculate $2ab$ which is $2 \times (3x) \times 4 = 24x$.

3. Calculate $b^2$ which is $4^2 = 16$.

Combine the results:

• $a^2 + 2ab + b^2 = 9x^2 + 24x + 16$.

Therefore, the expanded form of $(3x + 4)^2$ is $9x^2 + 24x + 16$.

The correct answer choice is: $9x^2 + 24x + 16$.

To solve the quadratic equation $4x^2 = 12x - 9$, we begin by rewriting it in standard quadratic form:

$4x^2 - 12x + 9 = 0$

Here, we compare to the general form $ax^2 + bx + c = 0$ and identify:

• $a = 4$
• $b = -12$
• $c = 9$

We will now use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute in the values for $a$, $b$, and $c$:

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Simplify:

$x = \frac{12 \pm \sqrt{144 - 144}}{8}$

$x = \frac{12 \pm \sqrt{0}}{8}$

$x = \frac{12 \pm 0}{8}$

This simplifies further to:

$x = \frac{12}{8} = \frac{3}{2}$

Therefore, the solution to the equation $4x^2 = 12x - 9$ is $x = \frac{3}{2}$.

Proceed to solve the given equation:

$x^2+10x=-25$

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

$x^2+10x=-25 \\ x^2+10x+25=0 \\$Note that the expression on the left side can be factored 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:

$25=5^2$

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

$x^2+10x+25=0 \\ \downarrow\\ \textcolor{red}{x}^2+10x+\textcolor{blue}{5}^2=0$

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:

$\textcolor{red}{x}^2+\underline{10x}+\textcolor{blue}{5}^2=0$

Notice that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{5}^2$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 (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 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 as:

$\textcolor{red}{x}^2+\underline{10x}+\textcolor{blue}{5}^2=0 \\ \updownarrow\text{?}\\ \textcolor{red}{x}^2+\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{5}}+\textcolor{blue}{5}^2=0$

And indeed it is true that:

$2\cdot x\cdot5=10x$

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

$\textcolor{red}{x}^2+2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{5}+\textcolor{blue}{5}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{5})^2=0$

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

$(x+5)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x+5=\pm0\\ x+5=0\\ \boxed{x=-5}$

Let's summarize the solution of the equation:

$x^2+10x=-25 \\ x^2+10x+25=0 \\ \downarrow\\ \textcolor{red}{x}^2+2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{5}+\textcolor{blue}{5}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{5})^2=0 \\ \downarrow\\ x+5=0\\ \downarrow\\ \boxed{x=-5}$

Therefore the correct answer is answer C.

In order to solve the exercise, we will need to know the abbreviated multiplication formula:

In this exercise, we will use the formula twice:

$(x+1)^2=x^2+2x+1$

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

Now, we add:

$x^2+2x+1+x^2+4x+4=2x^2+6x+5$

x²+2x+1+x²+4x+4=
2x²+6x+5

Note that a common factor can be extracted from part of the digits: $2(x^2+3x)+5$

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

• Step 1: Calculate each square.
• Step 2: Add these results.
• Step 3: Attempt to express it in a specific form, looking for a pattern based on known formulas.

Now, let's work through each step:

Step 1: Calculate the squared terms separately:
$2^2 = 4$
$3^2 = 9$

Step 2: Add these results along with the constant 12:
$4 + 12 + 9 = 25$

Step 3: Express 25 as a square of a sum.
Notice that $25 = (5)^2$.
We must check if this can be represented in the form $(a + b)^2 = a^2 + 2ab + b^2$.

The expression $(2+3)^2$ expands as follows:
$(2+3)^2 = 2^2 + 2 \cdot 2 \cdot 3 + 3^2 = 4 + 12 + 9$

The left-hand side $(2+3)^2$ perfectly matches our computed right-hand side $4 + 12 + 9$, verifying that this is correct.

Therefore, the expression can indeed be simplified as: $(2+3)^2$.

To solve this problem, we need to express $(a+3b)^2$ in two forms: as a multiplication of like terms and as an addition of polynomial terms.

• Step 1: Express as a multiplication
  The expression $(a+3b)^2$ can be rewritten as $(a+3b)(a+3b)$. This indicates the binomial is multiplied by itself.
• Step 2: Expand using the binomial theorem
  We apply the formula for squaring a binomial:
  $(x+y)^2 = x^2 + 2xy + y^2$.
  Here, let $x = a$ and $y = 3b$. Applying the formula we get:
  $(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$.
• Step 3: Simplify
  Perform the calculations for each term:
  • $a^2$: The square of $a$.
  • $2(a)(3b) = 6ab$: Multiply the terms and the coefficients.
  • $(3b)^2 = 9b^2$: Square the coefficient and the variable.
  Combining these, the expanded form is:
  $a^2 + 6ab + 9b^2$.

Thus, the expression as a multiplication is $(a+3b)(a+3b)$, and as an addition, it is $a^2 + 6ab + 9b^2$.

Therefore, the solution to the problem is $(a+3b)(a+3b)$ and $a^2+6ab+9b^2$.

To express the given expression $(7b+3z)(7b+3z)$ as a sum and a power, we will follow these steps:

• Step 1: Identify the components as a binomial expansion:
  Let $a = 7b$ and $b = 3z$.
• Step 2: Use the formula for the square of a binomial, which is $(a+b)^2 = a^2 + 2ab + b^2$.
• Step 3: Calculate each term:
• $a^2 = (7b)^2 = 49b^2$
• $b^2 = (3z)^2 = 9z^2$
• $2ab = 2(7b)(3z) = 42bz$

By substituting these into the formula, we get:
$(7b+3z)^2 = 49b^2 + 2(7b)(3z) + 9z^2$

Therefore, the expression as a sum is $49b^2 + 42bz + 9z^2$, and as a power, it is $(7b+3z)^2$.

Thus, the solution to the problem is:

$49b^2 + 42bz + 9z^2$

$(7b+3z)^2$

Proceed to solve the given equation:

$y^2+4y+2=-2$

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

$y^2+4y+2=-2 \\ y^2+4y+2+2=0 \\ y^2+4y+4=0$

Note that the expression on the left side can be factored 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 shown below:

$4=2^2$

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

$y^2+4y+4=0 \\ \downarrow\\ \textcolor{red}{y}^2+4y+\textcolor{blue}{2}^2=0$

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:

$\textcolor{red}{y}^2+\underline{4y}+\textcolor{blue}{2}^2=0$

Note that the terms $\textcolor{red}{y}^2,\hspace{6pt}\textcolor{blue}{2}^2$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):

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

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

$\textcolor{red}{y}^2+\underline{4y}+\textcolor{blue}{2}^2 =0 \\ \updownarrow\text{?}\\ \textcolor{red}{y}^2+\underline{2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{2}}+\textcolor{blue}{2}^2 =0$

And indeed it is true that:

$2\cdot y\cdot2=4y$

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

$\textcolor{red}{y}^2+2\cdot\textcolor{red}{y}\cdot\textcolor{blue}{2}+\textcolor{blue}{2}^2=0 \\ \downarrow\\ (\textcolor{red}{y}+\textcolor{blue}{2})^2=0$

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:

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

Let's summarize the solution of the equation:

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

Therefore the correct answer is answer D.

To solve the quadratic equation $x^2 + 32x = -256$, we will use the method of completing the square.

First, we rewrite the equation by moving all terms to one side:
$x^2 + 32x + 256 = 0$.

Next, we complete the square for the expression $x^2 + 32x$. We want to express it in the form $(x + a)^2$. To do this, take half of the coefficient of $x$ (which is 32), square it, and add and subtract the square inside the expression:
- Half of 32 is 16.
- Squaring 16 gives 256.
- Therefore, $x^2 + 32x = (x + 16)^2 - 256$.

Substitute back into the equation:
$(x + 16)^2 - 256 + 256 = 0$
which simplifies to $(x + 16)^2 = 0$.

To find $x$, solve the equation $(x + 16)^2 = 0$:
Taking the square root of both sides gives $x + 16 = 0$.
Thus, $x = -16$.

Therefore, the solution to the quadratic equation is $x = -16$.